Introduction to Analysis

Introduction to Analysis A'YIalysis· by

MAXWELL ROSEN,LICHcr ROSEJ\lLICHT VniveTsity BerJtele, Vnivmity of California at Bnl{eley

PUBLlCA'TIONS, !N.C. DOVER PUBLlCA~ONS, INC. NewTorl{ New Tori{

Copyript C CopyrIpt C> 1968 by Maxwen RoaenUcbt RoeenJiebt All riahta rilhta reaened under Pan Ameriean AD American and Intem&tional International Copyrtpt riIht OHlYentiona. Conventions. Published Publilhing Company, Ltd., 30 PubIiahed in Canada by General PublilhinK LeImiIl Road, Don Mills, Milia, 1bronto, LesmUl Toronto, Ontario. Published in the United Kingdom by Conatable Publlahed CoDItabJe and Company, Company. Ltd. 1bia Dover edition. edition, first ftr8t published in 1986. 1986, is This Dowr ill an unabridged unabridpd and unaltered republication of the work first tirstpubliahed Scott, ForeiForeSpubliahed by Scott. man and Company, CompanY. Glenview, Glenview. Illinois, Illinoil. in 1968. Arneriea Manufactured in the United States of Ameriea Dover Street, Mineola, N. Y. 11601 Dowr Publications, Publieationa. Inc., Im:•• 31 East 2nd Street. LIbrar7 fA Cata1o~ng.ln.PubileatioDData U......., ~ Conll'ftl Conlftll Catalo~ng-ln-Publlcatlon RoaeD1ieht, Maxwell Rc.ealieht. Maxwen. Introckaetion lJItroduetion to analysis. Reprint. Originally published: Glenview, IU. : Scott. Scott, Foresman, ReprInt. Glenview. In. Foresman. and Co., 1968. l888. Bibliography: p. BIbJioIraphy: lneladea lneludea index. I. Mathematical analysis. I. Title. QAaOO.R63 1986 516 86-25300 515 85-25300 ISBN 0-488-6&038-3 G-486-66038-S (pbk.)

Preface

This text is the outgrowth of a course given at Berkeley since 1960. The object is to redo calculus correctly in a setting of sufficient generality to provide a reasonable foundation for advanced work in various branches of analysis. The emphasis is 011 on abstraction, concreteness, and simplicity. A few abstract ideas are aln1<)8t minimal in number. llUDlber. Such u.re introduced, almost important concepts as metric nlctric space, conlpactness, compactness, and uniform convergence are discussed in such a manner that they will not need to be redone later. They are given concrete illustration and their worth is demoIlstrated demollstrated by using them to prove the results of calculus, generalized in ways that are obviously meaningful and practical. The background recommended is any first course in calculus, through partial differentiation and multiple integrals (although, as 88 a matter of fact, nothing is assumed a88umed except for the axioms of the real number system). A person completing most of the material in this book should not only have compreheD8ion of basic real analysis but should also be ready a respectable comprehension to take serious courses COUl'8ell in such subjects as integration theory, complex variable, differential equations, equatioD8, other topics in analysis, and general topology. Experience indicates that this material is acC688ible accessible to a wide range of students, including many with primary interests outside matheDlathestre88 on the easier problems. matics, provided there is a stress The quest for simplicity has resulted in the elimination of a host of omi88ion or relegation to the problem mathematical synonyms and the omission sets of a number of important ideas. Some problems, easily recognizable, &88ume a familiarity with linear algebra that was considered unwise to assume presume of all students. Indeed only a few simple facts on determinant. determinants are needed for the chapters on multivariable calculus, but things are Il1O 80 arranged that the instructor who wishes to avail himself of the conveniencee conveniences 80 with no break in continuity. Differential of linear algebra may easily do 80 forms were excluded, regrettably, to avoid exorbitant algebraic detours. This text can be used for courses ranging in length from one to t.wo two covered the first eight quarters. The original semester course at Berkeley coveled with the omission of most of the third section of the eiPth chapters, withthe eighth chapter. At present Berkeley has a two-quarter sequence with the first quarter, IIIOmewhat sped up, covering the first six chapters and parts of the seventh. somewhat 80me comments on the individual chapters. Chapter I, which Here are some 011 basic set theory thu.t that is familiar to many students, can discusses material on be covered very rapidly. Chapter II gives a brief account of how all the properties of the real numbers can be deduced from a few axioms. This

mat.crial material can also all10 be covered rapidly. It should not be byp888Cd, bypassed, however, for contrary to a widespread faith in modern pedagogy, my experience has been that time spent here is not wasted. Chapters III and IV, on metric spaces and continuous functions, are the meat of the book. They must be flow along done with great care. After this, Chapters V, VI, and VII ftow smoothly, for their substance (elementary calculus) is familiar and the proof. proofs now make sense. Chapter VIII, which is about existence theorems, is of a slightly greater order of difficulty. One may save some time by going very lightly over the implicit function theorem if it is intended to do the general case later, in the following chapter. The last section of the chapter treats ordinary differential equations, and the classroom discWllion discussion of these may with relative impunity be restricted to the very first theorem. Chapters IX and X, on multivariable calculus, conclude the book. They cause no difficulty for anyone who hu has come this far. However, should cauae omittiDl omitting them entirely would be prefer_ble preferable to an attempt to rush through. It is impossible to write a text such 88 as this without an obvious indebtedDeae Mlyai,. My Il'atitude ness to J. Dieudonnt1's Dieudonn6's classic Foundotitml Poundationa oj Modem M adem A Anal".,. gratitude is also due my colleagues of the curricular reform committee at Berkeley which instituted the new Mathematics 104 course, to Mrs. Sandra Cleveland for writing up my original lectures, to Adam Koranyi, whose revised notes have long been in use, for many conversations, to numerous other colleagues and students for their comments, to my family for its patience, and to Scott, Foresman and Company for its affable efficiency.

Berkeley, California

Maxwell Rosenlicht

Contents

CHAPTER I. NOTIONS FROM SET THEORY 11. II. Seta Sets and elements. Subsets Beta 12. Operations on sets 13. Functions 14. Finite and infinite sets

Problems Problema CHAPTER II. THE TIlE 11. 12. 13.

SYSTEM REAL NUMBER SYSTDI The field fteJd properties propertiea Order The least Ieut upper bound property 14. existence of square roots roote • 4. The exilltence Problema Pro~~

CHAPTER III. METRIC SPACES

11. Definition of metric apace. spaee. Examples ..

, 2. 12. 13. 14. ,• 5. 5. , 6. 16.

Open and closed seta Convergent Converpnt sequences aequencea CompJetenelll Completeness CompactDe8S Compactnell Connectedness Connectednell

Problema CHAPTER FUNCflON8 CIIAPTER IV. CONTINUOUS Ft1NCTIONS 11. Definition of continuity. Examp1et Examples 12. Continuity and limits 13. The continuity of rational operationl. operatiooa. PuacIioDI PuDctioaI with valuea in •r CoDtinUOUI functions on a compact met.ric 14. Continuous metric .... epue 15. CoDtinuoua funetione functions on I S. Continuous OD a 00DIleCted cormected .......... naetric epue 16. 8equeneea Sequenees of functions

Problems P~b~. CHAPTER V• V. DIFFERENTIATION DIFFEIIENTJATION defiDitiOD of derivative 11. The defUUtion dilerentiation 12. Ru1et Rules of of differentiation

ta. 13. 14.

'4.

The,IIlMIl value The.meaD value &Iaeore th80nlB TayIor'. ~ Taylor'. theorem. Problema

I

JJ 40 "

• 10 11

11 II 16 19 •II II 18 •29 II as It M 31 11 "44

11

.. " S9 It 61 6J. ",., .. 61 71

'I.

'II

.,. 'II • • .. •

I.

'. " •

_ _ 111

. . 1_ 1. 108

aL\PI'ER CIUPTER VI. RIEMANN INTEGRATION

11. II. Definitiona Definitiou and examples 12. Linearity and order properties of the integral 13. Existence of the integral 14. The fundamental theorem of or ealcuIua caleuIua 15. The logarithmic and exponential functions fUDctione

Problems alAPl'ER LIMIT OPERATIONS aIAPI'EIl VII. INTERCHANGE OF UMIT

tII. 1.

eequences Integration and differentiation ditterentiation of aequences

mN~~u of functions 12. Infinite IDfinite aeries I a. Power I8ries aeriea 13. funotioDII 14. The triIonometric tripnometric functioaa 16. DitrerentiatioD intecral lip OOfereatiation UDder under the intecral

~mu ~

CBAPrEB CILU'TBR VlD. VID. TIlE MBTIIOD OF SUCCESSIVE StJa:ESS1VE APPROXIMATIONS

11. The fixed point theorem

12. simplest cue of the implicit kDp6eit function theorem U. '!be TIle aim.pleat I a. ExiIkmce Exiet.ence and UIliquena 18. ~ tbeoremI theorems for ordinary cliIeNn&ial dilerential equatioDa equatioIIB

Problema ~

111 lIZ 111 116 118 123 lIS 128 III 132 131 137 117

1. lSI

1.1 1'1 ISO 156

159 119 I~ 1~

169 170 171 I71

177 190 1M

aIAPTBR CIIAPI'BR IX. PARTIAL DIFFERENTIA.TION DIFFERENTIATION DefiDitiona and basie bailie propertiee 11. DefiDitioDa p~ 12. JDcher derivatives t 2. JIiIher 13. The implieit implicit function theonm theorem

193 19S

Problema ~

IU

CILU'TBR X. MULTIPLE INTEGRALS CBAPrEB 11. BiImun . . . interval in ... Riemarm intep.tion iDtep'&tioD OD OIl . • . o1OIed and bailie propertiea _plea and· bulc properties ErIeteece of the in"",. ....tion OD 12. JbWeDce iatepal. lD latepation on arbitrary .. baeta of ... 1Ubeet.a P. Volume Vol\lDle

1M 194

.1 101

_-.s

III J11

,2.

116

,f. I..

lSI ISS lIS Mt M4t

13. Iterated intepala intepaJa 18. 0IIup of variable CIIup Problema

.1

III

8VCGBITION8 I'OR lOR FURTHER READING RUDING SUCGBI'I1ON8

U9 249

INDEX

lSI

Introduction to Analysis

CHAPTER 1I

Notions from' Set Theory

Bet theory is the languap of mathematic-. mMhematicl. Tbe Set ideas in modern are mOlt complicated ideu modem mathematies mathematiel . . developed in terms terma of the buio basic nodo. let theory• dev.oped notioDi of . . theory. . Fortunately the grammar.and vocabuluy of . ... theory pammar.and voeabuIarT . theol7 are exVemely extremely simple, at least in the .I8DIe . . . that it. poesimple. atleaat it • polaible to 10 very far in mathematic. mathematiClil ·with with 0D11 001,. ..a ema1I ema11 Bible set theory. It 80 so happens amount of let bappena that the subject aubj.& 01 let underlies mathematiee mathernatiee but ... baa ~ theory not only uncIerHes itlelf an extensive branch braneh of study; atudy; howev. however we become itle1f do not enter deeply into thia this study Itudy because beeauae there it ill DO need to. All we must is familiarise ouneIv. mUlt do here it some of the basic baaie ide8110 ideas 10 that tbelaquap the lan&uace ...., with eon1e mt.)' be 11Ied readilll of t.biI tbiI chapter can ueed with precision. A fint readiDi be very rapid linee is mainly a matter of .. ,tins UIId ainee it i. pttinc WIld t1 t directed worda. There i. oocuional verboli verboeity to a few words. ia occuional ideas whicb which toward the clarification of certain simple id.. they appear. &ppeU'. are really somewhat more subtle than the7

I

no..

THJ:OBY I. NOTIONS I'BOM 8ft IIIIT THlIOaT

§fl. 1. SETS AND ELEMENTS. SUBSETS.

We do not attempt to define the word.,. word ad. Intuitively a set is a collection, or ",gregale, Gfgregale, or family, famil'll, or ensemble en.semble (all of which words are used synonymoualy mously with let) set) of objects which are called the elementa, or member. of the set, and the set is completely determined by the knowledge of which objects are elements of it. We may speak, for example, of the set of students ItudentB at a eet are the individual individualatudenta certain. univenity; the elements of this thie set students there. numbers (to be diacuased discWJSed in Similarly we may speak lpeak of the set of all real numbel'l lOme allitraight lines in a given eome detail in the next chapter), or the set of aliitraight plane, etc. It should Ihould be noted that the elements of a set may themselves be sets; for example each element of the set of allitraight all straight lines in a given plane is less mathematical examples ia a set of points, and we may also alao consider such 1881 .. the let set of married couples in a given town, or the set of regiments in an army. We .hall seta and lower-cue lower-caae ehalI generally UIe use capital letters lettel'l to denote sets letten letters to denote their elements. elementa. The symbol Iymbol E is used to denote membership ehip in a let, set, 10 that

,nember.

~E8 zES

meana statement "z u% ia is not an element means that % z is an element of the set 8. S. The ltatement of 8" S" is abbreviated

z(£8. zf!S.

Inatead of writing a(J E S, be b E 8, S, c E Instead E S8 (the commu commas having the arne .me s. meaning .. u "and") we often write a, 4, b, c E 8. The statement atatement that a set Bet is completely determined by ita its elements may Yare be written .. &8 followa: If X and Y are sets seta then X .. Y if and only if, for z E Y. Equality here and elsewhere in thia all z, z E X if and only if % this book (denoted by -) means identity; X and Y happen to be different Iymbola, symbols, but they may very well be different names for the same set, in which case cue - Y means that the sets indicated by the symbols X aad the equation X =and same. X X" COUJ'le means that the sets indicated by the Y are the aame. " Y of coul'8e aymbola X and Y are not the same. symbols ThUi we imagine oUJ'lelves Thus ourselves in a world peopled by certain "objects" called "sets"), and for some pail'l (certain of which are caned pairs of objects z, X, wh. . X is a set, we write write:ex E X, the symbol E having the property that where EX lwo ... X and Y are equal if and only if for each object z~ we have z% E if and only if :Ie symbol E E mUit ~ E Y. The .,01001 must alao &110 bave have other properties (which dan't specify here) that enable UB, given certain sets, to construct otheJ'l. we don" others. tbina is that tbat everything which followa The important thina follows is expreBlible expreeaible in terms 01 the fUDclameDtai fundamental relation z E X. of ::I

_0 Ie.

tl._AllD .........

II

Seta are IOmetinuw IOmetimes indieated WOK III aD their members between iDdieated by Iiaciq br&cell. For example, the let _ braces.

(Jane, Jim} Jim" IJaoe, baa Jane and .nm Jim .. u ita memben, (1, 2,3, (1. 2, 3, ••. ) Uaod 10 ie the let _ of poeitive inteprB. (with the three dote .I'e8d . . "and 80 on") hi intelera, and

f-t fat'. _ havilll having one element, element. the object named G. _. (Note that f(a) G J ie •ia the let is not aame .... aame way there theN ia the 1&ID8 88 CI. In the same is a& ditferenee difference between a university clUl and the student himself, or between a com(,1)mcluB coDIiat.iDI coDIIiatiDg of one student atudent aod atudent hilDll8lf, mittee coosiating person.) The above notation how00DIIiat.i1ll of one person peraon and that peraon.) ever is not always feuihle. frequentJy used notation is feuible. A more frequendy z) J, Iz : (statement involving s)J,

(~

which JneUlI is true. meaoa the let aet of all sz for which the statement involving z~ ia Thus

is a a poeitive integerJ (s : z ie is intelera, and for any let ia the let aet of poeit,ive positive intepra, aet 8 we have

8-ls:sE81. 8 - fz: zE8). any let aet 8, the I)'IDboi aymboI For aDy (It&tement involviDg involvilll z») s)} (z fs E 8 : (atatement

deoot.deDotea the Mme -.me ..... set ..

(z (% : :i ZE 8 and (statement (8tatement involving z) %) Jt,, let which ia the . . of all elements of 8 for which the atatement statement ia is true. Thus . of real numbers, if R is the . let

(:l:ER: zI-l) -lJ. Iz E R : :r:Il' -== (1, -I). If X and Y are seta aeta and every element of X is al80 aIao an element of Y. Y, we Bay a.y that X is a ..." ...., of Y; this is written

XC Y, or Y:JX. Y:::>X. XC ie shorthand ahorthand for the statement atatement "if z~ E Thus X C Y i8 E X then s:I: E E Y". is equivalent to the two atatements X - Y ia statements X C Y and Y C X. x. IfU X C Yy aod Y C Z then clearly X C Z; z; the two &nst &Del first atatements statements are IOmetimea IOmetim. aucoinotly . written more IUccinctly

XcYCZ. XcyCZ.

",

NO'l'IONa J'BOII noll 81rr 811T TRIIORT I. NOTIONS THEORY

For any auy object z and set X the relation z E X can now be written in another (I. . convenient!) (~J eX. The negation of X XC (1_ convenient I) way, namely (zl C Y iI • written rt.. Y or YY~X. 1) X. Xtl,

x

X X ,. Y. X.iI called a fWO'IMr JWOPII' aubNt eubuI 0/ Y if XC Y but X,. The empC1I'" em", .., •is the set with no elementl. elements. It is denoted by the symbol (6. bepnnen •is that although t.he the empty let oonCOD91. A lOurce IOUI'Ce of confUlion confusion to beKinnel'l taine some particular let, the one ODe taint ftOCAiftf. noCIam" it iitle1f ....f is ~ing lOfMIAing (namely lOme characterised by the fact that nothing is in it.). it). The set. let (911 t f2J) •is a let. set confact. that. taininl eimilar way, when tainilll exactly one element, namely the empty set. (In a similar dealinl integers, we mUit. must be careful not dea1in1 with numben, numbers, .y lay with wit.h ordinary intepl'l, aero 81 nothing: 181'0 to reprd the number ero aero is something, a part.icular particular number, which repreaentl zero and 91 flJ repreeents the number of things thinp in "nothing". Thus 181'0 are quite different, but there is a connection between them in that the set let ~ 91 bu hu aero IeIO elements.) Note that for any set X we have

fllCX XCX. III C X and XC X. A special epecial cue of both of these statements is the statement

JlJ C Ill, (4, III occuiont difficulty diJlloulty If, iI, u is often improperly done, one ...... which oceuiODI reads 'Ije "Ie contained 9J C III flJ oontailled in" lor for both of the symbols symbole C and E. The statement III ia true tNe becaUle becalMe the statement "for each z E III II ~ we have z ~E E Ill" flJ" is obviouely obviously true, &Ild no z E III flJ and allO allO becaWle because it is "vacuously true", that is there is DO statement must for which the etatement mOlt be verified, just. just u 88 the statement "all pip apeak Chinese" Chineee" is vacuously true. with winp speak

I12. Z. OPERATIONS ON SETS. Yare If X and Y are aete, lets, the inter.edion inter,~tion 0/ oj X and Y, denoted by X ("\ " Y, is set of all objects which are both elements of X and eledefined to be the Bet ments of Y. In aymbols, eymbole, menta

X (\ ("\ Y .... X == (z Iz : z

YI. EX X and z E Y).

union oj oJ X and Y, denoted XU The tmion X V Y, is the Bet set of all objects which elements of at least leut one of the sets Bets X and Y. That i8. are elementAl is. XU X V Y is the set of all objects which are either elemente elementl of X or elements of Y Y (or of both), ineymboll in symbols XUYXU y =- (z:zEX t~ : ~ E X or zEYJ. % E YI. The word "or" Clor" is ueed uaed here in the manner that is standard in mathematica. mathematics. In ordinary language the word "or" is often exclusive, that is, if A and B are ltatements, statements, then IIU A or B" is underetood &1'8 understood to mean "A Ie A or B but not whereu in mathematics mathematice it altDtJlI' both", whereas alway, meane" means Ie A or B, or both A and B". BU.

t,2. 2.

....... 0 ... 011 .... ..... 0. . . .'110. . 0If

•I

. 8, then the eom""" ~ 0/ oJ X .. 8 ia If X ia ie a aubaet aubeet of a . let ie the . . 01 all e1ementa elementa of 8 which ..... DOt not elementa of X. If it ie ia expIieitly expIieltly 1tAted, Itated, or clear from the context, euctl)' ie, we often omit the wordI euctly what the set S 8 is. "in 8" and 11M the notation for the complement of X. TbU TbM iI is

ex fsEB: sEXI. sf! XI. ex - tsES:

Th . . operatione operations ..... illustrated in P'ipre Fipre 1, where the ... leta in queetioD Th. In quedioD seta of pointa in plane rePma repone bounded by curvee. curves. are eeta

xny xnY

ex

XUy XUY FrOUD 1. IDtM'IectioD,lIIIioD, Intenect.ion, union, aDd and oompIemen'oompIemenL Frouu

For another example, let 8 be the let set 01 of real numbenl, and aDd let

x-

y-

(sE8:s~ll, X - (s E 8: s ~ 11, Y - tsE8:0~sSII. (s E 8: 0 ~ s S 11. (The symbols eymbola ~ and S will be defined later.) Then

X/,,\Y- III Xf"\YXVY- (sE8:s~OI (sE8:s~OI XVY· eX-lzE8:s<-1I. eX - (sE8: s <'Il.

relationa hold among the symbols Certain relations eymbola A, V, e. For example, if Yare aubeeta of a& let set S, 8, then X and Y are eubeete eX f"\ A eY •- e(x e(X V V). This is illustrated in Figure 2. A proof of tbia fonnula is ia P\W civen below.

1111,

=,

HH.

FJOUIIII 2. eX Ie ilshaded ilshaded e(X v Y) lellhacleclliEl, illhadecl FroUJlll ehaded IIII , eY lelhaded mu.tIatiq iDuIf.ratIIIr ex 1"\ f'\ eY - e(X v y).

ex

n.

6•

•• NOTIONS J'IIOII nIOll 8ft SlIT TBIIOBY TllIIOBY I. NOTIONB

ExncIsB. ("\ e Y Y == = e(x v v V). Y). ExDcIs.. Prove that if xes, X C 8, Yes, Y C 8, then eX '"' It mud mlllt be he ahown that the two seta haYe have the .me _me elemeota, elementa, in other words worda that each e1emeDt aet on the left is element of the set ill an element of the set on the ript and vice vena. vera. If sz E ex "eY, f"I eY, theD then % z E eX and z E eYe eY. TbiI Tbia meana means that z E 8, z e e: X, ~·X U z E Y. Since % z e: X, %z ~ e: Y, we know that that. sz e:X V Y. Hence z E e(X U V V). Y). Conversely, if % z E e(X U V Y), then % z E 8 and % zE e: X U V Y. Therefore % zE e: X Convenely, e: Y. Thus ThIl8 z E eX and % z E eY, 10 that % z E eX f'\ f"I eYe eY. Thil Thill completes completee the and z E

e

proof. proal.

If X and Y are seta, the notation X - Y is sometimes BOmetimes used for (s EX: sx ft. f1. YI. Thus if X and Y are 8Ubseta BOme set 8, S, then Is subsets of some X - Y - X('\eY. X("\eY. Two seta are said to be diajoinl disjoint if they have no element in common. That is, X and Y are disjoint if X ("\ " Y = flJ. A collection of any number of. leta of.. seta is aid to be disjoint if every two of the sets are disjoint. seta may be defined in . The intersection and union of more than two sets seta then an obvious manner. For example, if X, Y, Z ar.e at:«' sets

X" X ("\ Y ('\ ("\ Z - Is Ix : % E X, sx E Y, 1Ia

2: x

E ZI,

and

XVYVZxu YVZ -

(x:sEX xEY xEZI. {~: zEX or zE Y or zEZI.

("\ yY ('\ ("\ Z =... (X" (X ("\ Y) (\ ("\ Z == X (\ ("\ (Y" (Y ("\ Z), and similarly for Clearly X X" the union of three seta. More generally the intenection intersection and union of arbitrary families of sets seta may be defined, and in an obvious way. The only seta, problem is finding an adequate notation for an arbitrary family of Beta, tbia is done .. follows. folloW1l. Let 1 be any set and lor for each i E 1 let Xi be and tlUl (80 that we may .peak whoee another set let (10 apeak of 1 as being an induing ~ng /4miI7l, famil" whose elementa are indica irtdit:u used to specify the sets seta at which we direct our main e1ementa attenQoo). aU seta Xi" X,88 i ranges over 1 is denoted . attention). The Bet set of all (X, : i E Ell (X,I., IX, 1) or IX.)., interaedion and union tbia family of sets, seta, together with their and the ifiWaectitm vnion of this

respective conventional symbols, are defined by

.,.,•,

(\X,(z:foreacbiEI,zEXil, ("\ X, - (x: for each i E 1, x E X,I,

vV X, =- Iz Xi'" (x : for at least one i E I, % zE

ie, «:1

Xi). X.I.

., .,

EXBRCIBB. Prove that if 1 and 8S are seta and if for each i E EXDCIB.. E 1 we have X, X 4 C 8, (eX,)• S, then e(f"'\ e«("\ X,) - U V (eX,). iEl

..................

iE'

It. __ Ibowa t.ba\ eacb element of the . let OD the left Ie ill an element 01 of the . let I' _ . he be ebowa that each t on t on the

,..... . . . . . ftnL

•'2.0~ftO.80N~ 2. OP:lJU.TlOJlB ON UTI

"X,)

7

"X,.

U f"I XI) then z E 8 and z E f"I XI. Therefore z E X" for H zE E e( fEI fOl' at leut least one .EI ~I lEI Thus z E eX" 10 that z E U (ex,). (ex,). i; E 1. Thua IEr 'EI Converaely, if z E U (eX.), fOl' lOme lOme;i E 1 we have z E eX" ex" Thua Thus z E 8 Convcnely, (eX,), then for lEI 'el and z E X,. Since z E X, we have z E f"I e(f"I XI). " X,. Therefore z E e(" X,). ThiI This completee comp\eta lei ~I

lei 'EI

the proof. tbeproof.

If G /I and b are objecta, objects, by the ordered ptJir (/I, (G, b) we mean the two objects G/I and b in a definite order, G/I first, b second. Thus if G, e, d are /I, b, c, objecta objects then (/I, (G, b) .. == (e, G -= ce and b - d. Note the distincobjecta (c, d) if and only if a tion between (a, (G, b) and (a, (G, bf; bl i the latter is a set with two elementa elements (unless, (unlefl8, COU1'8e, a G happens to equal b, li, in which cue case I(G, of coune, a, bIJ - (a I, a set with one hI can equally well be written (b, element), and la, bl /b, aI, spoiling the order. We remark that instead of introducing the new concept "ordered pair" terms of into set theory, we can actually define the ordered pair (a, b) in tenns sets that we already have: we set (a, b) .. ... the primitive notions about seta I(al, (G, b bll. II a I, (a, II. This definition does precisely what we want: to any two objecta objects a, b (di8tinct (distinct or not) it asaigns auigns an object (a, b), and it does this in tl) if and only if a .. 8uch such a fashion faeruon that (a, b) - (c, (e, d) ... ce and b .. =- d.

Given two sets seta X and Y we define the carteaian cartuian prod4ld product (or prodtACl) product) oj X and pairs the fint tmd Y, denoted X X Y, to be the set of all ordered paim fi1'8t member of which is i8 in X, the second in Y, that is

X X Y .. = (x, y) :zEX! : x E X, yEYI. y E YI. XX (z,y) uaual pictorial Ordinary rectangular coordinates in the plane give the usual representation of the cartesian product: the whole plane can be identified repreeentation niore with the product of the two coordinate axes. In Figure 3 there is a more complicated picture in which X, Y are subseta subsets of the two coordinate axes Yare ~baet of the first quadrant. and the cartesian product is a ~bset

•

•

II FJoU1lll FioUBII 3. Cartesian product.

8

t. I. NOTIONS FROM SET SilT THEORY

FUNCI10NS. 13. FUNCI'IONS.

junction Jrom Irom X If X and Yare sets, by a function Irom from X to Y (or a Junction into Y, or a function on X with "alue8 value8 in Y) is meant a rule which associates with each element of X a definite element of Y. (The word mapping, or map, is often used instead of function.) The "rule" can he be given in many waye, ways, some of which are discussed below, but the essential thing is that given any element of X there is associated, somehow, some definite element of Y. Two functions from X to Y are considered equal if and only if both functions 8880ciate associate with each specific element of X the same element of Y. Functions are u8ua11y emaIl letters, such 8uch as &8 J. The stateusually denoted by small ment HJ "J is a function from XX to Y" is often written

I: X --to Y. J:X-Y. For any ~a: E X, the element of Y that the function J associates 8880ciatee with z (the value of z) is denoted J(z). 01 J at :r) I(a:). Thus if J: X --+ Y and g: X --. Y then J -= g if and only if J(x) g(z) for all :e ,eMa a::e into 1I(z) .. - g(a:) a: E X. W~ say that J smds !(x), 2: and I(z) f(x) correspond corrupond under J. I. I(z), or that J I maps map8 za: into !(~), I (a:) , or that :r The rule defining a given function I: J: X --+ Y may be given in various ways. One way, which is usually not very practir.al, practical, is to list all ali the elements of X, listing with each one the corresponding one of Y. Or the rule may be

are both taken given by a mathematical formula. For example, if X and Y Yare to be the set R H of real numbers, an equation like

fez) = Zl + 3z J(x) = x' + 3x - 2 R ;i in such a case one often speaks defines a real-valued function J on H

+

(imprecisely I) of the function :r' xl + ax 3% -- 2. Again, if X is a subset of H, R, (imprecisely!) a real-valued function on X may be given geometrically by its graph, that (a:, I(z» : zx E X) XI in the plane; note that this method is the set of points I(~,f(x» mayor may not be practical, depending on what fJ is like, for it may not be p08sible possible to "draw" the graph. Rl'aph. In fact any subset of the plane defines a real-valued function on a subset of the real numbers, provided that any (z -= - constant) intersects the subset lubset of the plane in at most vertical line (x one point. Finally we remark that the "rule" defining a function need not practically computable. For example, for z% any real number, let I(a:) be practicaJly /(z) denote that integer 0, 1, ... , 9 which is in the billionth decimal place of :r J: precile, since a real number a:z may have more than one decimal (to be precise, == .9999 ... , we might better take J(a:) /(z) to representation, 88 in 1.0000 ... = he the smallest possible integer in the billionth decimal place of :r); :a:); this rule gives an honest-to-goodness function I: f: H R --+ H, R, but who would hazard 88 to the value of 1(11'), f(1r), or even 1(...;7J.)1 f(V2)1 a guess as Though this is in no way essential for what follows, we remark that it is easy to define the notion of function in terms of more primitive concepts set theory, as 88 follows: If X and Y Yare of sct are sets, a function from X into Y is a subset of X X X Y with the property that for any x z E X there is one and

13.

rt1NcnONI ruNCftONI

9

only one 1/ 11 E Y such that (%, (z, 1/) is in the subset. If the function is denoted

J: X ..... courae,/(%). analOlY - Y Y then the unique 11 1/ referred to above is, of COUl'lle, J(:e). In analOl)' with the case of real-valued functiona' functions' on the real numbers, this lubeet subset (I (%, J(%» : xz E X IJ of X X YY is called the graph of the function, 10 80 that (z, J(z» we are again saying that the graph detennines the function.

1 FrOUD

4. Graph of the function /: ~ R liven by I(~) /(~) I: R -+

Zl

for aU all sz E •. a. far

It is useful to note that the word "function" alone can be defined in not only the more complete concept "function flOm primitive terms, no't from X into Y": a function is an ordered pair whose first member ill is an ordered ordered. V), and whose second member is a function flOm pair of sets, say (X, Y), from IOmething of the type y), X into Y. That is, a function is 80nlething Y), (a certain y». This emphasizes that the I8tI kind of subset of X X Y». seta X and Y are to be considered /:'X --+ Y. For many purpu.... conaidered 88 &8 essential easential parts of the function J:X poI8I molt often we do not poses it is important to bear this fact in mind, but mOlt make any explicit mental note of it. For example, if I: X --. Y is a fUllCUon function and Y pt a function I': /': X -.... Y' by Y is a subset of another let set Y', then we set setting J'(z) J'(:r;) -= /(z) for all % ie, Ii vi nl I' the same II'&Ph If&Ph "I, u It - J(z) :e E X, that is, living il possible pouible since X X Y C X X X V'. which is Y'. Although I and I' /' &1'8 are really different functions, we uaually usually denote them by the same eam8 Iymbol, .ymbol, even expreeaion J: I: X --+ V'. Y'. In the same eame writing down the technically incorrect expl'ellion liven any function I: X -. - Y Yand and any subset 8ubset X" of X, we can de&ne define way, given - Y Y by I"(z) -/(%) -/(z) for all z:I: E a function 1": I": X" -+ E X"; I" is iI called the rearictioR 011 to to X". X". Of course, I" reslriction f" is not the same &8 88 I, though we olten often denote them by the same lame .ymhol, symbol, for example by writing the technically expl'ellion I: J: X" - Y when there is X" -+ i. no danger of confUlion, confuaio~ The incorrect expression graph of the latter function is of COUl'lle coune a& subeet subset of the graph of the oriPnai original function!: - Y, since X" X Y Y C X X Y. /: X -+ function J: X -+ and g: Y -+ functions, one can define the ctnrapoftctmpoaiIf !: - Y and,: - Z are functiona, lion oJ/and 01J and"g, or compoaed compoaedJunctioR, X into Z, tion Junction, a function flOm from X Z t by &8IOCiating &88OCiating to each element of X an element of Z in the obvious way: given an element fint uses J to get an element of Y, of X, one first Y J then one U8eI U8e8 , to get flOm from this

Y.

«X,

10

NOTlON8 nOM FROM SET 8ET THEORY I. NonON8

last element an element of Z. The composed function is usually denoted lut go J, 80 that we have 1,80 goJ: go/: X--+Z, X-Z, I) (x) = x E X. with (g 0 J)(%) == g(J(x» g(j(x» for each %

A function J: I: X ..... - Y is called one-to-one, or one-one, If if different elemente of X correspond under Ito I(z.) I to different elements of Y, that is if l(xl) 1(2.) ~I. X --+ Y is called onto if each element of I(~oi) only if Zl ~l - 2: •• A function I: X .orne element of X, that is if each 11y E Y is of the Y corresponds cornwpondl under 1 I to lOme form 'II -/(~), for some 11 -/(x), lOme x~ E X. If /: X X -+ - Y is both one-one and onto it is called one-one onto, or a onB-on6 one-one correapondence correapoodence between X and Y. An example of a function that is one-one onto is, for any set X, the identity lunction ix(x) === %2: for all x EX. E X. (Note that what junction ix: X -X, -+X, given by ir(x) x" in elenlentary elementary calculus is actually goes under the name "the function z" the identity function on the set of real numbers.) If I: f: X ---+ Y is one-one onto then each element of Y corresponds under I/ to one and only one elementof JI(y) === %ify == J(x). ment of X, 80 wecandefineafunctionj-l: we can define a function I-I: Y -+Xby - X by!-I(y) 2: if y = I (x). !-I ,,,,,erae Junction 01 oj I, J, and is also 0.180 one-one onto. Clearly /1 it called the inverae (Jl)-I .. and!-I 0 I === ix,! ix, lol""l = iF. iy. (JI)-I - J, I, and"'l 0 1 0 Jl == Y is a function and X' ex, If J: X -+ - Y C X, then the subset of Y given by J(X = I/(z) 1/(2:) : 2: I(X //) == ~ E X'}

(where the lut last 8ymbol symbol is shorthand for Iy such that I'll : there exists zx E X' 8uch 11y - /(z) J) is called the image I, or simply ainlply the image I(x)!) 'mtJ{/e of 01 X' under I, imtJ{/e 01 X', if . there is no danger of confusion. The two UBeI uses we have made for the .ymhol symbol 1 !(( ) are related by the equation 1(lz}) I«~J) - (/(s») II(~)J for each

% 2:

EX. E X.

If I: X X -. - Y is a function and Y' C Y, then the subset of X X given by

I-I(Y,) .. - (z Ix EX: I(z) I(x) E Y't yl} /-l(Y')

,,,,,erae

imOfe 01 Y' uruUr under f. I. It consists is called the inver. image consi8ts of those elemente elements of X which correspond under /I to elements of Y'. If I: X ---+ Y happens to be one-one onto we have another use for the symbol/-I, symbol I-I, namely the inverse function/-I: function!-I: Y -+ - X. These two uses of 1-1 I-I must be carefully distinguished, though confusion rarely arises. If I: X ---+ Y is olle-one Olle-Olle onto then the two uses of !-I fJ-l(y)!I -= I-I({y\) e&<~h y E Y. U8eI j-l are related by (J-l(l/) == I-I( (y J) for ea(~h

,4. FINITE AND INFINITE SETS. 14.

positive integer" integers, or natural numbels numbJa We are familiar with the set of poBitive 11, •.. }. This set, together with the various ideas associated with it, t1, 2, 3, ••. INCh .. as ita ite orderilll ordering (the fact that its elements can be written down in a IUCh de&Dlte de8Dlte ord.), order), or luch wch as the fact that two of ita elements may be added

14.

I'INITB AND INI'lNITII INftNlfl"B IIIITII 8JmJ I'lNITII

11

to obtain a third with certain general rule! rules holding for this addition, can be obtained from the primitive principles of set theory. In this text we shall instead assume &8Bume the basic buic properties of the real number system and from those derive all the properties of the set 11,2,3, 11, 2, 3, ... 1. In this section we shall facta about the natural numbe numbel"8 Bhall for convenience assume 8BBume a few simple facts ... in order to get as eaay matte... matters of set let u quickly as u pouible possible to certain other eBBy theory. However all the facts about the set of natural numbe numben ... that are used uaed here will be proved explicitly in the next chapter. The notions developed in this section will not be applied until later, so 80 no circular reuoning reasoning occurs. OCCUl'l. Let us therefore assume (I J 2, 3, ... 1. ). A set X is &8Bume knowledge of the set 11,2,3, called finite if it is empty or there is a positive integer n such that X can be put into one-one correspondence with the set 11,2 (I, 2,•... ... , nl, n) J that is there is a one-one function from (1, ... , nlonto n} .onto X. Thus a set is finite /1,22,•...• if we can count its elenlents elements and run out of elements after we count a certain number, say ft, rrhe number n depends only on the set aet X, not on n, of them. The the order in which its elelnellta otT j n is called the number 0/ of elements are counted off elementa elementB in X. For ~ompleteness oompleteness we say that the number of elements in the empty set is zero. Any subset of a finite set is itBelf enlpty itself finite, and if it is a proper hu a smaller nUlnber number of elements. subset it has 8ubset A set is infinite if it is not finite. This means that we can pick an element ZI E X, then an element ZI z, from the complement of I(ZI), EX. ZI), then an element XI cODlplement of (Xl, XII, etc etc., z. from the complement IZI. ZI). .• and we never run out of elements of X. Thus there exist distinct elements SI, Zl, Sa. ZI, SZI, ••• in X. ••••• eBBy to show that a set X is infinite infinite if and only if it may be put It is easy into one-one correspondence with a proper subset of itself. To do this, note first that if X is finite then any proper subset hu has a smaller number of elements. whereas two finite sets in one-one correspondence must have the ments, same number of elements. This proves the "if" part. On the other hand, if X is infinite then there exist distinct elements ZI, Xl, ZI. XI, Sa XI,•••• .••• in X. The complement of {Xl, IZI. XI, ZI, ZI, complenlent Xa, ••• 1 I in X is a subset Y. Y, so that

= {Xl, IZI. XI, ZI. X"~ ZI, X ==

·11r'\ (\ Y === flJ. ZI. ZI, A one-one correspondence between X and its proper subset I(XI, XI, ••• 1 ) U V •••

1 U Y tu

and

IZI. Sa. {Xl, Xt, Z" XI, ••• ••

Y is given by the function which sends each z. x. into Z.+l X.+l and each element of Y into itself. This proves the "only if" part. part, completing the proof. The set of natural numben nUlnbers can be used to give an eBBy easy definition of the notion of sequence. A 8equence sequence 0/ of n element8 elements in a 8et set X, or an n-'up16 n-tuple 0/ of elementB 0/ X, X. may be defined to be a function from 11, elBmenta (l, 2. 2, :.' ..• .. , nn)1into Xi X; ifthefunctionisdenoted/andwewrite/(l) /(n) - z., if the functionisdenotedjand wewritef(l) = == zl,/(2) xl,/(2) -.. Sa, Zt, •• •••••,/(",) X., (ZI, ZI then the n-tuple is often written (Xl, XI,•••• ••• ,, z.). x,,). An infinite 8equmc:e sequence 0/ of elements of 0/ X (or a 86qumc:6 sequence 0/ of elements elementB 0/ of X, if there isn't any danger of confusion with finite sequences) is a function from the set of all natural Xi &8 u above this can be written (ZI. numben into X; (ZI' Sa. ZI, ZI ZI,•••• ••• ). ), or. or, more conventionally. %1, ZI. Sa, sometimes Iz"I_.I •••.•.•• conventionally, %1, ZI XI,•••• ••• , or soluetimes tX" J.-1.1.1•.•• ell

11

I. JlfOftONl TRBORY JIIOI'rONlI PROII I'BOII 81ft' IIIIT TRIIORT

PROBLEMS

1. Let R be the set numbers and let the symbols symbola eat of or real numbera UoD&1 tlonal meaniDp. JllMlMp (a) Show that

S have their t.heir CODV8Il<, :S

. IzER:O;SzS3J"{zER: -1 (b) A U J2J - A V A - A "A - A U V flJ (e) f2J (0) A" A "flJ (d) A X fl1 flJ -

fiJ flJ

91. flJ.

8. of a eat set 8. Prove the foUowinl foUowiDlltatementa a. Let A, B, C 0 be eubletB IIlbeetll or etatementll and illustrate them with diagrams. diatrama. (a)
If A, B, C leta, show that U 0 are I8te, (a) (b) (c) (0) (d)

(A - B) ",,0 "0) - B C - (A "C) VB) V (B - A) (A V B) - (A "B) - (A - B) U 0) - (A :.. B) U V (A "B "0) A - (B - C) "C) XC 0 - (A x X C) 0) - (B X 0). (A - B) x

e

5. Let 1 be a nonempty set eat and for each 6. each,i e 1 let Xc X, be a set. Prove that (a) for any let eat B we have . (a> Bn U (B (BnX.) B" u V X~ XI - V "X.) 'er 'er 'el 'EI aubeet of or a given eat (b) if each X. is a subset set 8, then

e(fVeXi) XI)l l-e'el"r eX,. eX,. U

'EI

6.. 8•.

f: X ..... .... Y, ,: Y Prove that if I: y .... ..... Z, and 1a: A: z Z .... ..... W are functions, functiona, then "A •
n - (Ia • ,),> ·f· • J.

Letf:X .... Y 7. Let f: X ..... Y be a function, let A and B be IIlbeetll IUbeets or of X, and let 0 C and D be .. baete of or Y. Prove that aubaetll (a> I(A UB) VB) -/(A) U/(B) V/(B) "B) C/(A> C/(A) "/(B) (b) I(A nB) t-I(O V D) - 1-'(0) V J-l(D) t-I(D) (0) t-I(C '-'(C) U (d) ""(0" A/-I(D) 1-'(0" D) D> - rJ(C) r(O) "t-I(D) (e) ,-a(f(A»:> <el' ,..,.(I(A»:> A /(f't(0» C c O. (f) f(f-l(C»

c.

.

no.~ 11 ....... u I

8. Under the ..umpticma UlUJDptiona of Problem 7, prove that I/ is OftCM)D8 If U ud aDd . , If . . II oae-oae CJD1)o the be· repIaoed replaced by - for all A C:r, &Del I II is onto If if ud aacl ..., OIIIJ If lip :> in (e) can CUI be C X, aDd the lip in (I) aD be replaced by - for all aU C C Y. tip C ill (f) ...

I.

r. How IIIaDT manyaubeeta Bet (1,2, a, ... .. _,, til' A)? How IIIADJ maD,. .... JUPI of aubeete are there of the eat

tbia eet Row muymapl many'mapa of &hie this .. . . onto illelft it.lft eat into itlelf? itlelf' How 10. (a) How muy many IUDOtioDII functioDi are there from IlOIleIIlpty . . B iDto into 111' ~, flOlll a IlOIlfIIIlpty (b) Bow then fIJ into aD arbitrary arbitrarJ .. 81 How many lUIly functiona fuaotioDII are th . . from fIOIIll11 '" B? 8how that the notatioD notab It.I'd XII,., ...... (0) Show eEl implieitl7law1vt1 implicitly IIlYOlv. tIat the .,... aotIoa aU alfuaot.ioa.

CHAPTER II

The Real Number System

The real numbers 80 we muat must numbel'a are basic to analysis, 10 poMible to conhave a clear idea of what they are. It is pouible struct the real number system rigorous ayatem in an entirely rigoroua manner, startinl from careful atatementa ltatementl of a few of the buic principlea principles 01 fonow of let theory, • but we do not follow this approach here for lor two reuone. re&8ODI. One is that the detailed construction conetruction of the real number8, numben, while not very difficult, is time-consuming and fite fits more properly into a courae course on the foundations foundationa of arithmetic, and the other reason is that we already "know" the real realnumnumben bers and would like to get down to buaill8ll8. buainesa. On the other hand we have to be lure of what we are doing. aalUme cerOur procedure i.n this book is therefore to ClHUmI tain basio properties "(or axioma) of the real number (or axioms) .ystem, Iystem, all of which are in complete agreement with our intuition and all of which can be proved easily in the course COUl'l8 of any rigorous rigoroua construction of the .ystem. Iystem. We then sketch how moat most 01 of the falniliar properties of the real numbers are consequences of the baeic basic propthese propertiee properties actually comerties 818umed &8IUDl8d and how theBe detmnine the real numbera. numbers. The rest of the pletely determine coune will be built on this foundation. courae dartS procedure prooedure far • The ... ltandard ,Or OOIII~Uq CODItr~Uq the nal real DWIlben Dumbera II u folio.. : ODe followl: OM Int flnt . . buIe bulo . Nt. theory to de8De the llatural natural DUIDDumben tt1, 2, I, ••• , (wblob, to be&iD with, are mantly . witb 1, 8, 1 (wbloh, merely a . let with AD order relation), thea ODe de8Dea aD theD one defln. the additloA addltloA ADd aDd multlpllcamultiplicatloA of natural _ural DWIlben &Del abow. Dumben and aboWI that &heM theIe operatloAa operatioDa aatiIIy atlaly tion the famWar rulea of aIpbra. .. the naturaillWllben, familiar ruIee alpbra. UIi Uaiq natural DWDben, one thea theD de8n8II the .. to, :1::1, defines 88t, of iDt.epn inteprl 10, =1, :1::2, :i:2, ••• ,) &Del and exteDcia extenda tbe the opopoeratloAa of 01 addition and ADd multiplication to all the int.epra, eratiODl inteaera, apia ap1n verifyiq verifyinl the rulea 01 of alpbra. From the intepn integeR ODe one next obtailll obtaiDa Dumben, or fractioAl. FiDalIy,,from from the raticmal the ratioD&l rational DUmben, fractions. Finally ntional Ilwnben one 0118 ooutrllcta DUmbera conatructl the realilumben, real numben, the buic baaic idea in tbIa this Iut; step atep beinl bein, that a realilumber real number II ia aomethiq 80methinl that C&Il can be aplaat proximated arbitrarily cIoaeIy closely by ratlonailluJDben. rational Dumbera. (The maaumanufacture of the real Ililmbera number. may be witDeal8d witneased in E. Landall'. Landau',

beam

FtlUr&d4IitnM 01 of AftGlrN.) Foundationa Analrli•.)

16

II. TIlE TRIl REAL RJlAL NUMBER 8Y8TEM BYBTJIJI

, ]. THE FIELD PROPERTIES. 11. We define the real number .y.fem system to be a set Bet R together with an ordered pair of functions from R X R into R that lIatillfy satisfy the eeven seven propertiee properties lilted listed sections of thill this chapter. The elementll elements of R in this thill and the succeeding lIuccecding two eections are called real number" just number.. The two functions are called number., or jUllt addition and multiplication, and they make correepond correspond to an element (0, (a, b) E R X X R specific elements element! of R that are denoted by a + + bh and a • b respectively. rt!llJpectively. We speak system, rather than a real number eyetem, ayetem, IIpeak of the tAe real number llyetem, becauee listed propertit!IIJ properties thill chapter that the lilted because it will be shown at the end of this sense that if we have two completely determine the real numbers, in the eenee systems sets R can eyetems which satisfy our properties propertiee then the two underlying eete be put into a unique one-one correspondenoe correspondence in such a way that the functions + and • agree. Thus the basic assumption made in this chapter is that tJa system of real numbers exists. exist!. The five properties listed in this section are called the JW,d field properties becauee of the mathematical convention calling a field any set, together because with two functions + and ., properties. They expresa express the " satisfying these propertiee. fact that the real numbers are a field.

+

+

PROPERTY' PROPERTY I

I.

(COMMUTATIVITY).

+

For every a, b E R, V16 we have hove

a+b=b+a o + b == b (J and a·b 0 • b ... =t b·o. b • 0. For every G, 0, b, cc E R, tile tIM have Aav, (a + + b) + + c ... + (b + + c) and (G' (0 == a + (0· b) • c - atJ •• (h (b • c).

PROPERTY

II.

PROPERTY

III.

(DISTRIBUTIVITY). (DI8TRIBUTlVITY).

PROPERTY

IV.

or NEUTRAL ELEMENTS). (EXISTENCE OF

(AasOCIATIVITY). (AssOCIATIVITY).

For every a, 4, h, cc E R, a(J •• (b + c) - aG •• b + aG •• c. C.

tDf) !De

have

There are ore diatind di,tinct elementa 0 and 1t 0/ R IUCA elementl aucA that tAat lor /or all a E E R tile tH 1aaH lave a+O-acmda·t (J + 0 == a and a • 1 ... a. o. II:

PROPERTY

V.

or ADDITIVE AND MULTIPLICATIVE (EXISTENCE 01' MULTIPLICATlVm IHYEa_). INVZR8.).

For any (Ja E R there i. i, an on element 0/ oj R, denoI«I tknotetl -a, IUCA eucA that 0a + (-0) .... == 0, and /or for any non.nro t'ItOMero a G E R tI&er. 1Aer, iI if an element 0/ 0/ R, denoted denolMl a-I, 0-1, IUCA aucA that aG •• cr r 1l -- 1. Moat of the rules rulee of elementary algebra Most alp;ehra can be justified by these th~ five propertiee of the real number system. sy8teln. The main rnain coneequencee consequences of the field properties propertiee are given in paragraphs F 1 through FlO F 10 immediately below, properties together with brief demonstrations. We shall employ the common notational conventions of elementary algebra when no confusion ill is possible.

'1.

,1.

rmLD PmLD PIIOPIIIm.. PROP.."..

17

For example, we often write ab for a • b. One auch such convention is already implicit in the statement atatement of the distributive diatributive property (Property III above), is meaningless unless we know the order where the expression expJ'el!8ion a • b + 0a • c ia meaninglet!8 unlet!8 in which the varioUs performed, that is how parenth_ parenthelel various operations are to be perionned, ahould be inserted; by (Ia •• b + 0a •• c we of course mean (a (0 should (0 •• b) (o • c).

+

+

F 1. In a sum numbe18 parentheaea parentheeee can be aum or product of several real numberl omitted. That ia, is, the way parentheeee parentheaea are inserted is immaterial. Thus if a, b, C, E R, the expreaion 0 +b +c + c, ddEB, expreBBion a + d may be defined to be the eommon +o(b + c» c» +d +tl:ll + c) +cI+dcommon value of (0 +-(b - «0 + b) + (0 + 0 + (b + + (c + ••• ; b that (a + b) + (c + + d) - a + d» - "'; t these expresshown siona aions with parentheses parentheaea indeed poIIeI8 po88e8II a common value can be .hOWD by repeated application of the &8IOCiative fact aaaociative property. The pneral fact. (with perhape summands or factol'l) facto...) can be proved perhap8 more than four .ummanda by starting expreeeion involving element. of B, R, atarting with any meaningful expreBBion parentheaea, and several +'a parenthesel, +'8 or several ·'a, -'8, and repeatedly.hoving repeatedly.bovinl as many parentheses parentheaea as possible poesible all the way to the left, always ending up with an expreaion expJ'el!8ion of the type + b) + c) + d.

«0 «a

F 2.

In a 8um numbers the order of the tenna terms is aum or product of several real numbe1'8 immaterial. For example o-b·c==b·o-c==c-b·o== •••• a·b·c-b·a·c-c·b·a=- •.••

This Thia is ia shown by repeated application of the commutative property (together with F 1).

+

z +a R the equation :c 0 ... == b has hal one and only one 801uIOluF 3. For any a, b E B tion. For if ~z E B R is luch that z% + a0 - b, then z~ - zs + 0 ia such z + (a (-a» (z + a) :r: (0 + ( -0») lOS == (x 0) + (-0) .. == b (-a), (-0), 80 10 zZ - b + (-0) ia the only possible poaaible solution; 8Olution; that thia i8 this ia is indeed a 8Olution solution is immediate. One consequence is that the element 0 of Property IV is unique; another is that for any tJa E B, R, the element -a -G of Property V is unique.

+ +

+ +

+ +

+

For convenience, instead of b + (-a) -0) one usually writes b - Go o. (This is a of the aymbol"U -"between -" between two element. of B.) Go definition oftheeymbol e1ementaof R.) Thus - 0 - 0 --fl. We take the opportunity to reiterate here the important role 01 of cona + b + c haa has been defined (and by F 1 there iI vention. G i. only one I'8IIIOnreuonable way to define it), but we have not yet defined a G -- b - c. 01 Of course eoune expreaaion we understand und8l'8tand (a by the latter expreeeion (0 - b) - c, but it iI i8 important to realiae that this thia is ia merely convention, and reading aloud the worda "0 realise aufficient paUle pa.uae alter minus b minus c" with a sufficient after the fi1'8t first "minus" point. pointl out that our convention could equally well have defined 0CI - b - c to be standard convena0-- (b - c). In this connection note the absence of any Gn1lstandard tion for aa + b + c. In a similar connection, note that' that ~ could be taken (at)· if it were not conventionally taken to mean 0(11'). to mean (a6)e 0("). As atated ltated

+(

II

TIIII u.u. IfVlllll•• IYftIIII 8Y8!'1111 D. ftIII UAL ..VIllI••

above we ute UI8 aU the ordinary notational conventiona conventiOll8 when DO no confusion expresreeult. For example, without further ado we shall interpret an expreecan result. log 01 to mean 101 log (at) (01) and not 1.1)6, cab-I doee (cab)-1, lion like 100ai not. (1oe 0(111)', does not mean (ab)-I, etc.

I' 4. For an)' all)' a, R, with Ga .. ,. 0, the equaUon equation H_ -- b hu baa one F ,. CI, II E R. ODe and only one lOIutlon•.In _ - IIbfolio.. lOIution. In laat fACIt from .. foiloWi zZ -- :IGCI""' - bra, bcrl, and from •Z _,,-1 folio.. _ - II. - fHrl followt b. Th. ThUi the element 1 of Propen, Propert, IV II a ,. 0, the element er' unique and, liven any AD1 •a E ., a, ... crl of Property V is unique. it

..-1 -

be R, a '" ,. 0, we de&ne bIG, b/a, in aooold accord with convention, to be For 0, ClJ b E H, ....... ". cr. In particular, erl - 1/.. 1/ea.

.-1 -

0 E R we have a 0 • 0 - o. O. This ThiI it is tN. true linee G• 0 • 0 + CI a •0 F S. For any G a •• (0 + 0) - CIa •• 0 - •0 • 0 + 0, 10 10 that G0 • 0 and 0 are both lOlutioDi toIutiona G 3. From this thiI it of the equation ~Z + •0 ·• 0 - CI0 • 0, hence equal, by F 8. folloWi immediately that if •a product of IIveral aevera! elementl elementa of R 11 is 0 follow then one of the facton lacton must be 0: for if all cab - 0 and II a ,. ,. 0 .e we can erl to let I~ " II - O. Hence lhe the illegitimacy multiply both lid. by a-I nl.ti~ of divilion diviaion by aero.

-(-0) - . 0 for lor any. any 0 e E R. For both -(-o)-and anlOlutiona 1'6. -(-.) -(-a) "ADd 0G .... au'lou 01 of the equation s + (-a) bene, equal, by F 8. (-0) - 0, hence 3.

.-1 -

PT. (.... (cr-')-I an111OftIel'O 80 e E B. R. In fact alBae Binee 0CI • 0-1 - 1, I, by F 6 1''1. )-1 - •0 for &DylfOuero III faot we know that .... pi 0, 10 emt., cr-' ,. 10 (.-1)-& (cr-')-I . . . ., and F 4 impli. that (,.-1)-1 (crl)-I and 0 are an equal. equal, IIinoe aoIutiODl 01 equation za: • crt - 1. and. lin08 both are lOlutioDl of the' the'equation I'L Fa.

-(II + II) b) - (-a) lO1utioDl -(0 (-0) + (-b) for all aU a, 0, II E R. For both are aoIutions of 01 the equation za: + (a (0 + b) II) - o.

F9. (")-1 1". (cab)-' - .....-1 er'b-I if CI, 0,"b .... an IlOIlRlO IlOIlIel'O elementa elements 01 of R. For all,. cab,. 0 by F 5.10 5. 10 (.)-1 exiIta, 101Uti008 of the eData, and both (ab)-l (cab)-' and tr1bcr'b-I1 are 101utions equatioD equation z(ab) z(cab) - 1.

operatiq with fractions follow followeuily The UIual UIua1 rul. rut. for op.-tinI ..uy from FF9: 9:

o

ae fie -;;; --;;c

(ac)(bc)-I (ae)(be)-I - acb-1cr aeb-'c-'1 - cab-I -

c

ae (ab-I)(cd-II) - CIC(W)-I ac(btl)-, -liif' (alrl)(e:tt- : I

T· : • '7 ~ --

..!. +.!. _ ad .!. ""

"d

a

G bb' J

(ad)(bd)-I + (bc)(W)-l (bc){bd)-' + l!.. J!!.. _ (atl)(bd)-l

bel W btl btl be)(btl)-, _ ad ~ be • _ (ad + be)(W)-1

atJ,:!;

II. .... 12. OUIIII FlO.

+. - •. «

l'

19

+

-G -- (-1) (-1).• .forallae GforallGER. (-1).• G G+. - G' «-1) -G R. For (-1) -1) + 1) G aolutiona of the tbe equaCI •• 0 - 0, 80 that (-1) •• • G and -G -G are both lO1utioDl hence are equal. Two immediate conaequeD0e8 tion Z II - 0, henee co~ o· (-b) - II' G' IIb - (-0) are G· a- (-1) • "b - (-1) • II· (-a) • b - -ab and (-a) (-II) - -(0. (-6» - -(-ab) - abe (-0)'- (-b) -(0' (-b» abo

+

numbera, and therefore Notice that all five field properties properti. of the real numben, of them, are lMided aatiafied by the rational number8, numbera, or by the numberB. That is, the rational numbers and the complex numbers complex nwnben. are also alao fields. In fact there exist fields with only a finite number of e1ementa, elements, the simplest one beiDg being &a field with just the two elements 0 and 1. To describe the real numbers completely, more properties are needed.

aU all

CODIeqU8I1081 CODIIeQU8llC8

12. ORDER. 8y8tem is the following: The order pl'Operty property of the real number system PaoPIlRTY PBOPBRTY

VI.

TAtn ~ 01 R aucIa Ih.at '1' . .... ia it II au'-' Ilt IUCA IAtJt 0, 6 b E Ilt, ~, UNn tIun G a 1>, ~ (1) if a, b, a • b E Ilt CIA, aCI E R, H, one onl, OM one 01 1M elate(') lor G111I 0tN and GAd onl" tAl 10Uot.oing foUowiRf BItJlementa if ",.. . ." it",..

+

.e8+ .-0 -oE8+.

aEa. 0-0 -oEa.. The e1ement.l elementa G aE E R auoh IUOh that Ga E 8+ a. wiD will 01 00111II coune be called pont., poritiH, tboee lUoh IUch tbat 0 E R+ ~ ftfIItJIiH. negoliH. From the above property 01 of R+ we thole -CI ebaIl all the usual rul. rules for worldna worldng with inequaliti. inequalities. eball deduce aU . expre88 the coD8eQuenees consequences of Property VI ID08t moat conTo be able to .prell veniently we introduce >" and "<". U <". For tI, intmduce the relations relationa .. ">" a, 6b E B, R, either of the exprellliODB expreesiona a>b or b" btl and "I> Ub is respectively as &8 "a iii Iell Jell than aU) a") will mean that aII - 6I> E E Ilt. ~. Eidler Either of the exprellliOD8 expre88iona , a~b lI~b

bSa or 6Sa

will mean that 0II > bI> or G 0 - b. Clearly aG E a. ~ if and only if G R is negative if a > O. An element G0 E R a < O. and only if tI following are the CODleQUeDces consequences of the order property. The foUowiDa :III

(TrIoIaotomy). If II fI, a, b E E R then one and only one of the followiDa following 01. (TrIohotomy). ltatementl ltatementa ia is true: G >6 0>1>

.-b a-I>

o
10

n.

D. TIDD RJDAL NUMB• • 8T8T11)1 BT8TIJ1I TBII JtII'\L NUMIIIlR (I - b For if we apply part (2) of the order property to the number II then exactly one of three poasibilities G - Ii b - 0, poeaibilities holde, holda, G(I - b E R+, R+. (I or b - fI(I E 14, R+, which are the three C8H8 caBe8 of the 8I8ertion 811ertion 0 1.

(I > > band b > > c then II (I > > c. For we are given 02. (Transitivity). If II (J (I - b E 14 ~ and b - c E 14; ~ j it therefore fonows follows that 110 II (I > C. c. a(I -- cC - (0 - b) + (b - c) E R+, 10

+

03. If (J(I > > band c ~ d then a(J + Cc > > b + d. In fact, the hypothesee hypotheses a-bER+, c-dE~VIO), consequence mean II - bE R+, c - dE 14 U {OJ, and &II u a conaequence «(I + c) - (b + tl) d) - (0 - b) + (c ~, proving the .-ertion. ueertion. (0 (0 - d) tl) E 14, > 0 (meaning that 0a> > 0) and c ~ dtl > > 0, then 04. If 0a> >b> > band b > tIC > GC > btl. bd. For a(J -- b E R+ and c E 14, ~, 80 110 ac GC - be - (0 «(I - b)c E 14, ~, d E 14 ~ V 10) b E 14 ~ together imply that and similarly c - de to) and be bd E 14 ~ V 10) neceaearily folloW! follows that be - ME to};j it DeceB8&rily ac 14, that i. GC - btl bd .. - (ae (GC - be) + (be - btl) bd) E E~, i8 fJC GC > > btl. bd. 888Umptiona that b and tl d are positive are e8IeIltial; e88eDtial j the Note that the 888UmptioDl usertion doee not hold, for example, with a-I, uaertion 04 does 0 - 1, b - -1, c - 2, tl-3. fl- -8.

o S. 5.

The fonowing following rules of sign for adding and multiplying real numbel'l numbers hold: (positive number) + (positive number) .. - (positive number) (negative number) + (negative number) - (neptive (negative number) (positive number) • (positive number) - (positive number) (positive number) • (negative number) - (negative number) (negative number) • (negative number) - (positive number). (neptive These are immediate from F 10 and Property VI.

06. For any a(I E R we have ,.. al ~ 0, with the equality holding only if a(J .. general1y the IUJn IJUJIl of the squ.... squares of leveral eevera1 elemental element. 01 of 0; more generally R is always alW&)'I greater than or equal to 181'0, sera, with equality only if all the eJementl sero. For by 00, ltat.ement G element. in question are 181'0. 0 0, the statement 0 ,. .. 0 > 0, and a BUm IJUJIl of positive elementll elements is positive. Note the implies a ll > II poeitive. 1 > O. apecial coDllequence consequence 1 - 111> apeoial 07. If 0> tJ > 0, then 1/0 l/a > > O. In fact a· o' (I/o) (1/0) - 1 > > 0, which would rules of sign 1/(1 SO. contradict the rulel lign if we had 1/0

b> l/a < lIb. l/b. For ab Gb > > 0, hence (ab)-l (tJb)-l > > 0, 08. If a> tI > b > 0, then 1/0 (Gb)-Ib, which simplifies l/b> > (ab)-Ib, (ab)-IO limplifis to 1/" > l/a. I/o. a>

110 10

,2.0RDD 12.0RDD

11

09. We now show how the computational rules of elementary arithmetic as consequences of our Ulumptiona. aMunlptiolUl. Let UI make the defiwork out aa nitions 2 .. == 1 + 1, 3 -=- 2 + 1, 4 - 3 + 1, etc., and let UI us define the number, to be the set ). Since 1 > 0 it follow followa natural number. Bet 11,2,3, ... J. that 0 < 1 < 2 < 3 < .. ....'. The let of natural natunJ numbera is ie ordered exactly aa &8 we would like it to be-in particular, the natural numbera a, b, have the following properties: for any natural numben 0, b. exactly G, ", II, c are natural one of the statements atJ < b, 0 - h, b, bII < 0G holds; hoIdl; if a, numbers and a <" < band allO 0 < C; Cj any natural Dumber and b < c then aIao number baa h.. an immediate lucceeeor lucceuor (i. (it leut least natural number that II ia peater numbers have ditlerent 1Uethan it); different natural numben different immediate 100ceeeors; and there i. Ce880I1J; is a natural number 1 with the property that any set a1Io Bet of natural numbers that includes 1 and with each element aIao its ita immediate succeeeor BUcceseor coDliste consista of all natural numbers. numben. For any n, fa n il that ie one-one natural number ft, ie the sum of a set Bet of 1'1 I'. t,bat; i8 in ontM)ne co~pondence with the elements of the set (1, 2, 3, •.. "I. This correspondence •.• , ",. elementl of a let of implies that in whatever order we count off the elements is, a set in one-one correspondence correepondence with the set let nft objects (that ii, (1,2,3, n I) we arrive at the final count n, 11, 2, 3, ... , n) ft, and if a proper subeet sublet Bet of n objects haa 11& uaua1 rut. rules for of a set m objects, then 11& m < n. The U8U&I addilll numbera come from luch computations computationa .. aa adding natural numbers

+

+

+

.,.ter

+ 3 .. 2+ - (1

+ 1) + (1 + 1 + 1) - 1 + 1 + 1 + 1 + 1 -

6,

while the rules for multiplication follow from the fact that lums luma &8 products; productl; for example, for any of equal terms terma may be written aa a E R we "have Sca. Thus have G a + a + a0 .. - (1 + 1 + 1) • aG -- Sa. ThUi 3· 4 44: + 4 + 44: .. - 12, 10 we can verify the entire multiplication table, .. high .. .nteg••, that is ie tbe (0, =':1, =':2, u we care to go. The integ"." the IUbIet IUbiet 10,·::i: 1, :A:2, :3, R, are also &110 ordered in the correct way •.• ..• < -2 < =3, .. '. IJ of H, -1 < 0 < 1 < 2 < .... easy to cheek intepnl add .. .. It il is euy check that the intepn according to the ordinary rules; that they multiply in the U8U&I uauaI way i. is implied by FlO F 10 and the corresponding fact for lor the natural numraltonal number., ie the elements of R H whicb number" that is which can be bers. The rational written alb, with 0, integers and b.,. a, b integen b ~ 0, are aIao a1Io ordered in the usual way; indeed the order relation of two rational numbera numben can be determined by writing the two numbera numbers with a positive common numerators. Addition and multidenominator and comparing the numeratol8. numbers are aIao IUD8 0""'operaplication of rational numben a1Io determined detennined by the lURe tiona numbers, a oertain tione for lor the integers. integere. ThUi Thus the rational numbeN, certain IUbiet aubeet of R, propertiee with wblch H, have all the arithmetic and order properties wblcb we are familiar.

+ +

+ +

w.

Here is M aa good a place as aa any to introduce into our 10Iical lQlical ~on di8cuIIion of the real Dumber number system the notion of exponentiation with integral intepal exponents. If a E Rand Hand n is some positive integer we define 0- u» to be

Do . . . . . . . . I n J I I S D _

•

(ft timee), a_ • 0a ••...• • a ... a (n timel), and if •a ,. 0 we define .. - I, .a- - 1/.-. From theBe rules of exponentiation, tbeee definitions we immediately derive the uaual usual ruts

..... ........ --.-...........

in particular

....

(...))a_.. (.).-

The de&nltlon de&nltloa of the abeolute abIoIute value of or a ... lnOIIt conreal number is moet point.: if •a E R, the eNol.." of a, denoted veniently introduced at tbia point: CIIMolute lHIluc lHJlue 0/., 1.1, pven by 101, is aiven by

101- 1_11_1t-I- 0 1.1t-t- -(1) (2)

(3) (3) (4) (6)

if _ • > 0, if _ - 0, if 0<0. • < O.

The abaolute abeolute value baa the followiq following properties: propertis: tal ~ 0 for"'lo for alia E R, and 1_1tat - 0 if and only if •_ - 0 1.1 lobl-I.H"I E R tabl-Iat for all tI," a, 6 E I_II - .. forall_ER I_I' for all CI E R ICI + 61s lal + 161 forall.,bER for alia, 6 E R 1-+"lsl-I+lbl ICI61~ 11_1-1" 1101-1611II for aIlo,6ER. I_ - "I~ aIIo,bE R.

.'lI'

properties above ..... The lilt &rat three propertiea are trivial COD88qU8llC8l couaequencea of the definition at of I-I. I_I. To prove (4) note &rat tint. that :1:_ SI-I :1:Siol

I_I) :1:6 S S 1'1, ,.,. :1:"

(_ing (mMninl that _ S I_I and -0 -CI S ,at) and

:1:(0

+ b) S 1.1 + tbl,

or

lci+&lst-I+llIl. 10 +"1 SItli + Ibl. plOve (6), To prove (5), note t.bat.1al-l(a that.101-1(0 -- 6) b) + 61 bl s S ,_ Itl - III bl +161,10 + Ibl, 10 that t•.,lol-Ibl· ta.'- bl 61 ~ 1_1-161· Inteft:han&iDI _ and b, Inteftlbanging tI and fI, if;,)'

.,i,

l~ - bl III ~ 161-1_1. i~ Ibl-lol, the Jut two inequalities combine into (5). and th.1ut (6). gives It It ueful to note that repeated application of (4) givs

1_+_+ ... +10.1. 1-+-+ .. · +o.ISloll+lo,l+·" +ca.ISla.I+I ..I+ .. · +1a..1.

,a........ I 3.

.... LI1d'l' v U• • • HUJlD BOUND nonaTr PJIOPBaTY

JI D

We also uaeful fact that if z, z,o, alao note the trivial but very ueful 0, e E R, then

Iz - al < eE I:.: -01 if and only if (I-e
. . .---1---..1·--. ---"'1 •- -...,

1I..' - - - • .. t e - - - • - - - · ·... I I

I

I I

I

I

I

a-' l'Iouu I'IQUBII

a I. The Tbe poIna poIDtullllGh ,_ - _, 5. z auch that Iii G I < e.

G+'

At the end of the previous aection eection a& number of other BYStem. systems were liven satisfy the first five properties of the real number ay.tem. system. The liven which .tiafy order property excludes colllisting exeludes two of the systems given there: the field coDllisting of just the two elements elementa 0 and 1 (since then 1 + 1 -.. 0, contradicting 1 1 > 0), and the complex numbers (since any number must have a numbers .tiafy nonneptive equare). But the rational Dumbers satisfy all the properties &iven 10 far. Since Sinoe it is Ie known that there aiat liven exist real numbera numbers which are not (tbia will wiD be proved shortly), ehortly), .till rational (this still more properties are needed to d.mbe the real numbers completely. d.cribe

+

+

,So

I S. TIlE THE LEAST UPPER BOUND PROPERTY.

To introduce the lut fundamental property of the real number Dumber system Sy8tem fonowina: concepti. If 8 C R, then em we need the following Clft UPfl'l' upper bound/or botmtl lor at. tAB ,., Nt 8 Ie a number CIa E R such IUch that. S ~G a lor each •, E 11 E 8. If the aet let 8 baa hu &11 an upper we.y bourwIetl/rom bound, we .y that 8 Ie is bounded from GboH. abotJB. We call a real number 11 a , . " upper UPfI'I' botmtl bound of 0/ 1M at. .., ,." aet 8 if "it &11 upper bound for 8 (1) II is an and Ie any upper bound for 8, then 11 S ~ G. (2) if G is this definitioD definition it follows that two least upper bounds of a set From thia muat be I1eB8 . . than or equal to each other, hence equal. Thus a set 8 C R must aet 8 C R can have at most one least leut upper bound and we may apeak of at. tAe leut upper bound of 8 (if one exists). exiata). Note alao the following important least

14

n. U.

TO UAL HUMalUl NUMBI:R 8T8TlJM TIIII HAL BTIITIIM

fact: if 1/ls 11 is the leut fJ, then there ate exists least upper bound of 8 and z E R, z2: < 1/, an element element.8 E 8 such lOch that z% < 8. •. A nonempty finite subset 8 C R always haa has a least upper bound bound;j in this thl. cue the leut least upper bound is simply the greatest element of 8. More generally any sub8et has a greatest element (usually denoted IUbset 8 C R that haa max S) u a least upper bound. But an infinite IUbset subset of R need 8) hu haa max 8 aa not have a leut it8elf baa no upper bound least upper bound, for example, R iteelf at all. Furthermore, if a subset 8 S of R has does not Dot haa a least upper bound it dOM necessarily necesaarily follow that this least upper bound is in 88;j for example, if 8 is the set of. has no greatest element, but any of all negative numbers then 8 haa a ~ 0 is an upper bound of 8 and zero (a number noe not in 8) is the least upper bound of 8. The lut system il is the following, which gives laat axiom for the real number ayetem a further condition on the ordering of Property VI. PROPERTY PaOPDTY

ftOnemply "' ad 0/ A raonemply real number, is botmckd botmtl«d /rom from aboH 1&tJI 1uJa a l«ut l«Jat upper number. thtU tIuJl i.

VII.

(LBAST (LJlA8T UPPER BOUND PROPBRTY).

bound.

If we look at the real numbers geometrically, imagining them plotted on a straight line in the usual usual· manner of analytic geometry, Property VII becomes quite plausible. For if 8 C R is nonempty nonempty' and bounded from baa a greatest element or, if we try to pick a point in 8 above then either 8 hal uaa far to the right lUI poeeible, we can find a point in 8 luch such that no DO point rilht aa il more than a di8tance diltance of one unit to the right of the choeen in 8 is chOlen point. Then we can pick a point in 8 farther to the right than the first fint choeen chosen lOch that no point in 8 is il more than one-half unit to the right point and such of this second chosen point, then a point of 8 ltill still farther to the rilht right lOch that no point of 8 is more than one-third unit to the right of the lut such last choeen point, etc. It is intuitively clear that the sequence of choeen chosen choaen pointe in 8 S must "gang up" toward some point of R, and this last lut point will be the least upper bound of 8. (See Figure 6.) Another way to justify Property VII in our minds is to look upon the numbem 88 u represented by infinite decimals, i.e., symboll real numbers symbols of the form (inteRer) (integer)

+ .fIttJIfII .(ltGtOa ••• ,

aI, tit, «It, Ga, fit, ••• is one of the integers 0, 1,2, where each of the symbols (11, 1, 2, ... ,9, , 9, >. +. with the symbols <. <, >, +, .· being interpreted for infinite decimals decirnala in the standard way. (Note that any terminatinl decimal can be coneidered conlidered an aD

-1

o

2

a3

,I ,I I" II I IIII

FrQ1J1UI 8. A eequence aequence of 01 points pointe in R pngiq FIoUlUl6. pnKin. up toward, a Ieut upper bound. bounel.

13. ,3.

LZAST LZAIIT unz. UPPZR BOUND PROpaRTY PROPRRTT

15 25

infinite decimal by adding an infinite string of zeros.) If 8 is a nonempty set of infinite decimals that is bounded from above, then we can find an element of 8 whose integral part is maximal, then an element of 8 havinl having having the same integral part and with al (11 maximal, then an element of 8 haviDi the same integral part and 8&me 01 with tit same al CIt maximal, and we can continue this process proCeBIJ indefinitely, ending up with an infinite decimal (which may 8) which is clearly a leut 8.·• or may not be in S) least upper bound of 8. subset 8 of R will be denoted l.u.b. 8; The least upper bound of a 8ubset another common notation is sup 8S (sup standing for the Latin ",prem"",). IUprlm"m). Property VII says that l.u.b. 8 exists whenever 8 C R is nonempty and ConveJ'8ely, if 8 C R and l.u.b. 8S exista, exists, then 8 must bounded from above. Convel'lely, I18t and be nonempty (for any real number is an upper bound for the empty let there is no least real number) and bounded from above. Analogous to the above there are the notions notioll8 of lower bound and greatest lower bound: a (I E R is a lo1Der bound for the Bubeet subset 88 C R if G CI S , for each each,B E 8, greate8llower bound 0/ of 8S if (Ia is a lower bound of 8 S, and aa is a greoteBllo1Der and there exists no larger one. 8 is called bounded from below bek>w if it hu has a lower bound. It follows from Property VII that every set 8 of real numben that is nonempty and bounded from below has a greatest lower bound: as a matter of fact, a set 8 C R is bounded from below if and only if the let set S' == (z : -x E SJ 8' = Ix: 81 is bounded from above, and if 8S is nonempty and bounded from below then -l.u.b. 8' S' is the greatest lower bound of 8. The gre&te8t greatest lower bound of a subset 8S of R is denoted g.l.b. g.1.b. 8; another notation is inf 88 (inf abbreviating the Latin infimum). If S has a smallest element (for example, if S is finite and nonempty) then g.l.b. S is simply this 8 i8 smallest snlallest element, often denoted min S. consequences of Property VII. Among other We proceed to draw some eonsequenCe8 numben are not very far from the things we shall show that the real numbers IJenlJe that any real number may be "approximated rational numbers, in the sense as closely as numben. The way to view the situation 88 &8 we wish" by rational numbers. i8 is that the rational numben are in many ways very nice, but there are certain "gaps" among them that may prevent ua us from doing all the tbinp thinp as eolving equatioDi equatioDl (e.I., (e.g., extractinl extracting we would like to do with numben, such uBOlving roots), or measuring geometric objects, and the introduction of the real numbers numbeJ'8 that are not rational amounts to closing the gape. Here are the consequences coll8equences of the least upper bound property: Intepnll known, topther addldoD • Let UI remark here that once the .... let of lote..n II Imown, top~ with their adcll"and multiplication, it It i. II poIIible pouible to oonltruct t.he the real number ayatem eyatem by cIe8nlq de8DIna real numben AN numbel"l by mean. 01 of infini~ In&nlte deeim..... deelmall. Thia TbIa it II in fact the way real Dumben numberl . . uauaUy uaua1Iy It. II to eompute oompute with decldeciintroduced in elementary arithmetic, and we bow how euy it. mal•. But there are •a few ineonvenieneea inconveniencee in thll fact. that, that. mals. thia method atemmlnl ltemminl from the fut. lOme numben reprNeDtation (e.g., .999 .9119 ••• - 1.000 ••• ). number8 have more than one decimal representation II .110 allO the esthetic inconvenience of liml IiYinI •a preferred .tatus .tatUi to the DUmber number 10There is almost lhall discUlllater thil eeetion almOllt a biological biololical accident. In any cue caee we ehan dilCUil later in thia _tion how the seven leven properties of real numben imply that they can indeed be repreaeoted repreeented by In&aite in&Dite decimal8, decimal., thu8 thull eompleting completinl the circle with elementary arithmetic.

•

D. TIIII TIm IUIAL UAL NUIIBD HVIIBU 8YftDI 1'fftIIII

LUB 1.

For any real Dumber integer fa such IUch tha\ that 11 z. number s, there is an integer" R > s. (In other words, there exist arbitrarily large integen.) integers.) To prove this, _ume &Illume we have a real number sz for which the 8IIeI'tion &ll8ertion is wrolll. inteler ft, the let let, of intepra inteprl wronc. Then ft RS :s; s for lor each intepr R, 10 that \he it let of integen is nonemp\y nonempty iI bounded from above. Since Sinee the set it hal bM aA1. . upper bound, lay integer R, ft, R ft ...y .. But for any inte&er 1 fa aIIo alIO an in"" intepr, 10 "R + 1 S R :s; it ~• • and thUi 11 S •CI -- 1, I, ahowinl Ibowin. that G -- 1 Ia aIIo let 01 of inteprl. Bince that. Il1o aD upper bound for the set ., G• ia DOt not aAleut •CI -- 1 < fI, leut upper bound. This TbiI ia is A a conU8diction. contradiction.

+

LUB I. 2.

For any integer" such aDy positive reM real Dumber number • there exists an integer Rauch l/R < .. (In other words, there are arbitrarily small ama1l posithat 11ft tive rational numben.) For the proof it auflic. to choose an auflicee \0 integer" > lIe, U88 08, II., which is poeeible poeaible by LUB 1, then use which ie we, have 11. 1/e > o. O. iI permilBible peranilaible lince linee by 07 we

LUB 3.

For any zs E R there is an integer 1& such that R A :s; S sz < "ft + 1. aD intepr " luch To prove thie, - N < a: z < N. this, choose an integer intepr N > Is II,, 80 that -N intepn from -N to \0 N form the finite let (-N, The inteprl ( -N, -N + 1, I, •.• 0, 1, .•• take n to peateat ... ,,0,1, ... , N NtJ &Del aDd all we need do iI is take" \0 be the greatest of theIe ~. . these that ia is 1_ than or equal to \0 s.

LUB 4.

For any za: E R and positive integer N, there Ia II aD an intepr "n lUeh that such !!..~ !!.. .... < ,,+1 R+l N N;;::s N "' show thiI thia we merely have to \0 apply LUB 3 to \0 the number To ahow pttilllaD integer,. lUeh that that" :s; Nz Ns <" Ns, pttiDl an integer ft IUch " S < " + 1.

So LUB S.

If s, t• E R, • > 0, then there exist. exilfa aA rational number, Dumber, aueh luch that IIs ~ - r'1I <.. (In other worda, a A real number may be M clolely elolely . ...we approximated .. .e wiIh wim by a& rational number.) To this, u. 11M LUB 2 to find & a positive inteler N such IUch that prove tWa, <.,t, then UIe use LUB ..4: to find an integer liN < inteler " lUeh such that "IN S :s; Sa: < (A (,. + 1)/N. nlN < liN <"e, 10 so ",IN 1)IN. Then 0O:S; S sa: - ",IN lIN <

Is -ta/NI -RINI < t.•. I_

w. now dilc-. diacUil the decimal rep.... NpNleDt.ation tation of real numbers. numben. Firat Fint oonaid.. lnit. &nite decimala. U If ... Ia any intepr, " any positive integer, and lid. fie ia Cla,", , ... any aDy inteprl intepra choIen choeeo from Iymbol -it Ga, •••• ...... flOm amonl 0, 1,2, I, 2, ... ..• ,9, , 9, the .ymbol

..............a.

..... tIt •••

win mean, mean," usual, the rational number wiD .. UIU&1, CI. Gt

CIt

CIa a.

ao+lO+ 10·· "'+10+ 10' + ... + 10-'

13.

I&UT U"O VPPU IIOVIID BOVill) no.JIIlTY norllllTY 1&UT

17

If m is a positive integer less leu than ft, n, then (Io.al· •••• a. S ~ tJe.tIl (10.41 ••• a. a. == (Io.al cae.Ol. ao.0l-• •• - • a. ~ (Io.al" S CJe.Gl •• •.0. CIa

+ Cla+a CIwo+l • 10-<_+1) 10-<00+1) + . a. ·• Ur ..... + a. lO10-<00+1) + ·_. ... + 9· l(r. + 9 • 10-<_+1) 9 • 10-.

If we add 1010...... to this last number a lot of cancellation OCCunJ, OCCUl'I, resulting in (10.41 ••- •-Ga 0. S ~ (Io.al • • a. GeeGi. GteGI-• •• Ga

< Gt-Gt (10.41 ••• •• •ea. 0. + 10'-. 10--.

Thia Jut laat inequality is at the hue bue of DlO8t moet roundlng-otl rounding-otr procedures in approximate caloulations calculations and in addition shOWl &bow8 that two n~bera numbera in the poBBible addition of a above decimal form are equal only if (except for the possible DUmber seroa to the right, which doesn't change the value of the Iymbol) 8ymbol) number of seroe eame digits in corresponding places. It a1Io aI80 enables us to tell they have the same at a glance which of two numbers in the given form is larger. The ordinary lilIes for adding and multiplying numberB numbera in this thi8 form are clearly legitimate. roles inftnm tkcimal By an infinite decimal we mean a formal expression (Io.a.asaa ••• ao.01GlGa

(this is just another way of writing a sequence) where Go is an integer and ai, tit, III, I, ... ,9. The let each of (II, Ga, ••• -.. is one of the integers 0, 1, ((Io.al a. : f& n - positive integert (ao.OI ••• Ga integer! is nonempty and bounded from above integer m > 0, ae.41 Clg.al ••• 0. a. 10-10- is an upper bound) hence baa a& (for any inteaer leut upper bound. The 8ymbol Qe.G1GIfJa apamion for (Io.GllJa1i1.••••'. is called a decimal dBc1mal "pamion thilleut upper bound and we I&y eay that the leaat least upper bound is ,qreunted thill_t ia ,. .am" br the infinite decimal. Thus every infinite decimal ill is a decimal expanaion expansion decima.l itae1f it.lelf aa a for a definite real number and we may use the infinite decimal symbol for the number. Thus

+

Qo,a.asaa ••• CIoeG1GIGa•••

inteprt, l.u.b. ((Io.al."a. lao-GI ••• Ga : "n - positive inteprl,

and for any positive integer n we have the inequality (Io.al • • • a. ~ ao.01C1tCJa. ao.allJalil •.• ~ 00.01 Go.al ••• •• • a. Qo.41 ••• CIa S •• S CIa

10-. + 10--.

This enables us to tell immediately which of two infinite decimals represents the larger real number. Note that two ditTerent different infinite deciolals decimala may be decimal expansions for the same eame real number, for example exanlple 5.1399999 ... - 5.1400000 ... ,J but the last inequality shows 8hows that different 6.1399999 decimale are decimal expansioDl expansions for the earne infinite decimala aame real number only in thiI this cue, that is when we can let get one infinite decimal decima.l from the other by I, ... ,8 replacing one of the digits 0, 1, , 8 followed by an infinite sequence of nines by the next higher digit followed by a sequence of 1er08. leros. Din. decima.l. To Any real number is represented by at leut least one infinite decimal. 8ee this, apply LUB 4 to the cue 10"', where m is any poaitive positive integer: lee case N ... a: 10·, we get a finite decimal decima.l ao.Ol ••• •• • 0. luch that (Io.al ••• a. tJo.Gl •••

S Zz < CJo.Ol ao.al ••• a. + 10--. 10-. :S

18 •

II. TIIJI ftl!lAL 1roM••• !roMB• • STemM STIITSM IJ. TIIII .mAL

If we try doing this for m + 1 in place of m, then ao and the digiti digits ai, Gt, ... • •• ,, a". fJ,.. Il00+1. Letting m get larger will not change, and we simply get another digit tJ.+l. and larger, we get more and more digit. digits of an infinite decimal, aDd and this i. is our desired decimal expansion for~. Note that the addition or multiplication of two infinite decimals goee goes according to the usual rules: we round oft off each decimal and add or multiply the corresponding finite decimals to get desired sum or product. We obtain u many •a decimal approximation of the d.ired digiti AI u we wiah of the decimal expansion of the sum or product by rounding digit. 01 the given infinite decimal. decimals to a sufficiently large number of pl&eel. places. expansions of real numbers, it is very euy easy to exhibit Using decimal expallliolll teal real numben numbers which are not rational. One such number i• i•

.•101001000100001000001. 1010010001‫סס‬OO10‫סס‬oo1. .. . Multiply this by any positive integer and one gets a number which is not an integer, 80 this number cannot be rational. f 4. THE EXISTENCE OF SQUARE ROOTS. 14.

It itt is convenient to prove here a special result, even though this can be derived &8 a consequence of a much more general theorem to be proved later. A square root of a given number is a number whose square is the given number. Since the square of any nonzero number ill is positive, only nonroots. The number zero bu one square negative numbers can have square roote. root, which i. lero iteelf.

U 0 < Sa If XI < ZI z. then Zll ZI' < »1. ~. That iI, is, bigger positive numbers have bigger Iquarel. squares. ThUl any given real number can have at most one positive poeitive biaer Iquare has at least remains to show that if (J a E R, atJ > 0, then G a hu square root. It remaina one positive 8quare square root. For this purpose consider the Bet set 8 E R : z~ ~ 0, ~ B =a - (:,; {~E

~ aJ. Sal.

Thia set is nonempty, since.O E S, B, and bounded from above, since if This X > max (a, /a, 1) I} we have zI ~ -=- z Z •z Z >% x •• 1I == ~x > a. y -= ... I.u.b. l.u.b. 8B % d. Hence 11 exist.. We proceed to show that" - a. First, "y > 0, for min {I, al E E 8, B, exia•. fl, oJ since (min (1, (1,0»' /1,a} any.• such Bince 0)1 S min {l,a} (I, a) •·11 - min (1, at S a, tI. Next, for any lUoh 110 that 0 < • < "y we have 0 < JIy - • < 11y < 'IIy + e, 10 (y - .)1 e)' (,I

< 11' (y + ,)1, e)', " < (JI

positive numbere numbers have biller bigger squares. By the definition of 11y since bigger biller poaitive B, but 11y + • B. Again using the there are numbers greater than 11y - e• in S, e E 8. numbers have bigger squares, we get fact that bigger positive numben (y - e>' e)1

< IIa < (" (y + e)l.

PlIO. . . . . . . .

Hence

I". (y - .)1 I,. - al cal < (y + .)' e)' -
4,. 1_

by chOOling IImall enough we can make 4,1e I. . than any pree chOO8ing •e Ilmall PM' iped rip«! I,. - cal positive number. ThUll I". al ill 1_ I. . than any poaitive positive number. Since 1111 - al 01 ~ 0, we must have 111' 01 - 0, proving 11' IY' IlIl -- (II " -- a.

..,;a;

a > 0, the unique positive Ilquare IIquare root of • ill If (I is denoted ..,;-.; thua thus hu exactly two square -..."ra. W. We aIeo write lIquare roots, namely ya and -V-. &leo wriM V1f -0. y1f - O. aquaNI We now know that the positive real numbera numbers are precilely preciaely the IIqUU'el politi. . nnumberl ...... of the nonsero real numbers. Thill ,how ahowe that the .. . tt of poIitive completely determlntd determined R+ whose exilltence ill affirmed by Property VI ill is compieMly by the multiplication function of R. A priori, it mipt IMID eeem tha. that there then pcaible subaets IUbsets R+ of R for which Properti .. VI aDd could be several poIlIlible Propertiee aod VII dilleUllllion of the ordering of R the .......... hold and that in any diecUllllion IlUbeet ... .,... WCN1d unneeeaaary. The . . R, B, have to be specified, apecified, but we now know this to be unneceesary. together with the functions functioDII + and ., " determine the ordering oreleriDi of R. B. It ...... taa.. fore follow follows that the decimal expansions R are comple&e17 completely expanaiODll of e1ementl element. of B (B, Sine. the addition and multiplieatlon determined by the triple (R, Since mulijplication of decimals • real "ft....... . , . , . iI .. decimale follow the UIluai usual rules rul.. of arithmetic, 1M ...... .,.,. completely IeDIle that if we have compleUly thtmnin«l tlUrmined by Propertiu ProptJr'iu I-VII, in the IeDII8 bave (B', IIotillfying theee .. then there wiD 8IdIR satisfying these properti properties will . . another triple (R', a unique ontH)ne R' ...-villi pnlIIlI'Yil1lllUmI uad on~ne correspondence between R B and B' IUIDI and product.. ThUll we may speak of the tAe real number 1)'Stem. products. eyatem. In fact one ofteo of. speaks ie llpeaks of "the real numbers R", meaning the real number.,..; number .,aleMj this ill IItrictly speaking erroneous, erroneoUII, since aince B . and we allO strictly R ill merely a . let aleo bave have to know what the operatioll8 this set are, but when there then .. DO operatiollll + and • on thiII llno danger of confusion thill ill a convenient abbreviation.

va

o CI

+, .,. .).

." +', .')

PROBLEMS PROBUNS 1. Show that there exIatI mltl em. one and (.-ntlall,) WeI with three th........... (. . .tIalI)') oaI1 oaI)' ODe cmelleld ...... ca, 6, e, "• e R 2. Prove in detail that for any 0, (a) - (a (G -") - 6) - 6 G " - a II) - (ae (b) (G (0 - b)(e - Ii)

<M + be). + lid) W) -
3. Prove that if 0, b E R and 0G < 6 < 0, thea tIleD 1/. 1/0 > 1/6. lib. G, 6 4. (a) 11223/71 Ie 223/71 greater than 22/71 (b) Ie 285/163 266/163 peater areater thaD 1361/7801

•

D. '!'1m NUIDD 81Wn11 U. 'I'I1II UAL IIUID_ ft8'I'IIII

I. For which ~s E R are the following inequalities true? true' foUowiDc iDequalities (a) 3(s+2)<s+6 3(~ + 2) < s + 6 (b) s' ~-k-6~O - &:e - 6 ~ 0 (0)

!e > zs-1 - 1

(d)

>S+8>O. -L. ~ + 3 > o. 3> a- a s-

~

t. if a, 6, ., a, ,r e R aad aDd ° • < < r, << ., I" - _I << 6 - a. G. a. 8bow Show that lbat If < •• < < 6, • < 6, then Ir 7. &bow Show that for aay IUIY a,. 0, 6 e R,

1G -- 61 max 10,61 a + 6 +2 +21a max IG. 61 -_ G+ _mul_a,_61._a+6-2Ia-bl. min 10,61IG.61 - -maxI-G. -61- G+ 6 -21G - 61, 8. The . .pIa II IIUtn6er ia deflDecl (eaUed the The..apla ...... . ...... . II de&Ded to be the .t C - R X B (caDed . .pia ........ II. . . . .)) k»pt.her functioDa from C X C into C, de...". topther with the two fUDOtioDa DO&ecl by + + aDd "• t dlat, tllat are amm + (e, + e, DOted liven by (0, (a, 6) + (c, 4) d) - (a + c, 6 + + cI) 4) aDd (0, ') • (c, (e, fI) cI) - (ac (e - W, ., GIl + Ie) (a, 6c) for aU all ., a, "t, c, 0, " 4 e .. IL (a) Show that C, toptt. topther with the functio. and "., II ia a field. fuDctioaI + aDd (b) Show iIlto c I8DCIa -.ell aCI E B R into (0, (a, 0) II ia Sbow that the map from. R B into C which . . -.ch ~ ADd UId "PNIerftI .~ n I IIddit.IoD ODe-ODe MlclitloD aad aDd IIlIIltipUaatioD" multiplication" (beiq
°

ar •

Ja the IUbiat bouaded from above or below? no. I. Ie _beet liS /lJ of • bouacIed Does it have lUI aD I.u.b. or aa,.I.b.? •.l.b.?

I'IDd the tbe ,J.b. aDd La.b. ar the tbe 10UowiDs lolIcnrbaa ... 10. Ibacl IJ.b. &Del I.u.b. 01 .tat alvill, pviDl reuoua fIUOoa If if you (a) (a> (b)

{I,I,l,i, ... } {1.I.l.1.,..} {l,~,~,~,....} ...} U.~.~.:: {v'J.

CUI. caD.

+ J'i, v' + V2 +

"2+v'2+v'I, ... }. (0) {VI,v'2+J1, V2 2 v'l, ...}. 11. > 1, thea the eet' la,",", (a,", tI', ... ... 1 , II is not bounded from n. Prove that ifIf aa E ft,. B, 0> (BiN: Pint lint find &ad a positive poeitive intepr " such IUch lbat above. (Him: that G > 1 + ~ and prove

+

that .. > W

(1 +~r ~ 2.)

au'"

12. 1M X aDd aad Y be DODfIIIlP'Y 11. DODeIDpty 1Ubeet. ar of R B whole whose wdon union II is R and IIIlCb such tllat that -.ch -.ell ...... aI Z X II ... thaa each element aI e1eIIleatt of of Y. Prove that there exiata exists aCI E •R . .. . that: that Z X 18 II ODe 01 aI the two . . . leta I. or ,. < al. Is e •B :: •s S 01 oj Is e EB R : •s< at.

D", are DODelDpt)' _pty IUbeete of R lbat IL If Sa. S. .... _beeta 01 that are bo1lllded bounded from above, prove that tba' La·b.I. Lu.b. Is + 1/r :. : Z e 8" B1,,,r e Btl -l.u.b. - l.u.b. 8, 8 1 + I.u.b. l.u.b. Bt. St.

....... J'8OUoD8 I~. Ij.

11 II

• IlWDber Dumber S . tlW Let at CI, 6 • E B, with •CI < II. •. Show that, that. there exIata exiata Ii II e EB B. aueh that. • < •II < "fI, with s ratioDal wiIh. rat.ioDal or DOt ratioDal, .. we 1riIh.

16. "A real Dumber Is is rational if and baa a periodic decimal expaoaion." exp&IIIIion." 11. aDd only if it bas statemenf.. Define the ,.-eDt p .....nt ueap ueaae of the word pariod1c periodic and prove the 8tatement.

eapIoIIIiona of 01 real DWDberI Dumbers were deftoed by b), apeolal 18. Deolma1 Deolmal (1G-Dary) exp&DIioDl wen deftDed apeoia1 referDumber 10. Show that real Dumbers bave b-Dary b-rwy expanaioDl with eace to the DUlDber numben have aDaIopaI propenieI, w wben . .·....ter thaD IoD&lopua .... •" II ia lAy an)' iIl. iIltepr"poeatar than 1.

CHAPTER III

Metric Spaces

Mod ill con.-ned Moat. of elementary ana1yIia. anaIyaia iI concemecl with functioDl of one or more • ftmctioDl funetione more real variabl., variablee, that that. functions 01' the plane, plaae, or defined on a subeet eubeel of the real Une, or ordinary 3-epace 3-epaee or, more pneraIIy, pneraJly, tHtim_oul tHtimeneioul Une, plane, etc. are ..,... Euclidean space. apace. The real line, IJ)ICial c... II c. . . of the pnera1 concept of "meViclpaCf/' "meVic tpaCe" wWob which iI 00ft_· introduced in tIM tm. chapter. We W. a1Io alto introduce. conv_ent venient pometric 1anpap laquap for deali... d-.1iDl witb with ...ned "topolocical" queatioaa, queationa, which are queetioM queetiona . . . .W &ed "topololical" ..... with the notion of "pointe Dear DIU' eaeb eaoh fotber'.. I~', •• IlOtloa DOt1on ia a priori ratite rather vape. The ideu cleYelop ean that i•• W- we develop. . be applied to &81 any meUio metric epaee apaee at aD. ThilaotODlJ This _ oob' pIGpIOvidea a peat great economy of thouPt. thought, IIince will be .... ..... vid. Iiace it wm aary of ha'rial I&l'Y to introduce. introduce a Dew new id. idea only once"iDItead once inItead of-viOl to define it for each apecial _ that mA1 b1d lpeCial cue tbat IDA1 OCCV 0CCUl',t "'" will a1IO &lao lead to productive new ...,. 'Wa7I of looldDl looIdas .. at familiar objeetl. ~aIl objecte. ThUl one ".Dale lingle proof wiD .... eonr·aIl eateI limultaneouely and a .-tam . .., . appIIedla C8lellimuitaneously certain . ..-1\ . . . III •a manner IUpt)y ha........ aliptly out of the orcliDu7, ordiDu7, .., mat ..... ..... unapec$ed aad and far..rel.chilll far...reachina ~ •u will be UDexpecW eeen later when other exarDpIee eumpa. of JDekia metric .........

on

po"'"

M l . IlI:T1UC M i lIU. IOTIUC "SPACES ...CII8

11. DEFINmON DEFINITION OF METRIC SPACE. EXAMPLES.

A metric apace iI UIOCiatei with each ia &a let topther with a& rule which UIOClatei pair 01 element. .... number Dumber IUOh lUeb that certain axiom. axiOIDl are_tIIare _tilelemente of 01 the let a& .... fled. chOleD in IUch is ..-onable fleel. The axioll1l axiom. are chc.eD auch a manner that it la nuooabl. to think 01 the .. lett .. as a "apace" (a word we cIon't don't bother to define in iIolation), iIolatiOD), the element. number 8880Ciated wiUl two elamente of the set aet .. "pointa", "pointe", and the real nUmber UIOCiated wiUa elements of the set .. 88 the "diat&nce "diatance between two 'wo pointe". Here is ia the preciae precise definition:

w.

~don.. _ B, IHjiR' t Ion. A metric Ipaee ~ is a & let E, together with a rule which UIOCiatea 8880Ciatei with each pair p, fEE q) luch that I E B a real number d(p, f) (1) d(p, ,) ~ 0 for all 'I, EB p, ,IE (2) d(p, q) f) - 0 if and only if p - If (3) d(p, q) f) .. - d(q, d(f, p) for lor all p, , E E E B + tl(Q, d(f, r) for all p, (4) d(p, r) S d(p, f) q) + P, f. I, rEB r E E (triangle inequality).

a let and d Thua a metric space apace is an aD ordered pair (B, 11), cl), where B ia ila ThUl B X B ... .... B a function d: E R l&tiIfyiq latiafyiDI properti. (1)-(4). In dealiq dea1iq with a• metric apace (8, II) tI) i\ often undentood lrom from tbe context oontex, what wW d tJ la, or it II olten that a certain apecific apeeiflc dII II to be born. borne in mind, and then one often apeab mekic apace B"; this thia is II IoPcallY limply of "the metric 10IicaUy incorrect but but, Vert VerJ con con-.. venien'- The elementl elamente p".', p, I, r, ... of a• metric IPM8 venient. apace B (to be be· ahlolutelJ ablolute!y co~ we welhould BAY "the elements. of the underlyiq corne' Ihould .y uthe elemente.of underlyilll' Bet set 8B of the metric apace (8, (8,4)", ..... d}", but let 111 us not be too pedantic) pedaDtic) are called the pointe poi"" 01 of B, and if p, ,fEB d(p, ,) f) the diaItJnce anel E B we call tl(p, ~ lHrCtDem ~p P_ tmd fj 9; d II itaell itlelf is called tlitIGrtce /undioft, /vrtI:litm, or tIIAIIric. the tIitlaftce fIUIric. ap&ceI: Here are lOme exampl. examp1. of metric 8J)IW8: 0'0

w.

B -- R . (the let of'" of reU numben), d(p, f) (1) E q) -I, -11' - fl. 91. The 8mt three metric apace epace axiom. uiome obviously hold. The fourth foUowa follows from the computation d(p, r)

+ (, +

-I p - rl -I (p (P -

+

,) q) + (9 - r) I Sip - 91 qt + 1 IIq d(q, r). - d(p, q) f) + d("

rl

&Il1 pci.itive pci.ttive intepr integer " we define a metric apace (2) For aD1 space B·, E", called

Hi~ Bucli4eaft let of 1WIi~ BvclitleGA apace, ~, by takiq takil1l the underlyiq underlyillll8t 01 B" g" to be all numbera (Ga, ••• ,t cr.) B and definiDl, ft-tuplel of real numberl (GI, ••• a.) : Ga, GI, ••• ••• ,, cr. a.. E E R), deflnilll, !or .for

,....plea

", .. <-I, ···,aa), If -0 . 0 ' . .) ,

d(p, ,) f} -_ d{JJ,

I,

(f" WI, • •o.•,,.,.), II.) ,

~

0

+ <st -

+ ·.. + (z. -

Vr.(a:-I---.,-:.):-=-I-:-+-:(a:.~_-"':-::I-:+••-,-::+-=(;..-_-.,'7:.)1,

V (Sa -

11.)1

tlill

1Ia)l.

11. ,1.

DD1lOT1011 01' 0 .. . IdTIDO IPACII JI8IIII1'I'IOII . . .0 UAmI

II

We mUd .... actuall, .. a metric apace. .,ace. The irM DUId pI'Oft tba' tbM .. II '''''IT flrIt three metric apace uioma trivial to ftrify. verify, 10 it remainII remaiDs to prove the trian&le f,riaqIe uiomI are U'8 t.riYill inequality. We need two preliminary neultl. reeultl.

Propoadoa "",). 'ar 'or _, ,.., reed.umber. ai, a., CIt, ...• • •• , a.., Ga, Propoatdoa (SeA...... ........,.".). ""'" CIt, "., lit. ......... Ac. lit, lis, • • •, fl., .. -tOlllt + Gab. + + ..."-, -IGtbt Ga6.+ ·.. •",-+~""'~I~_~ :S ..; V,-O-ll-+-01 V btl bl l + bI ba' + ... ··· + bal. b.l , Cltl + 01....+-·-·-.-++ ... + ...-1.1 ..; The proof atarta (I, fJ E R we have ItartI with the remark that for any a, fJb,.)1 + (CIlIt (ClOt - fJb,)1 ... + (aa. fJb.)1 fJbJ)1 ~I + ·.. (era. - !Jb.)1 1) -- 2«~«(Jlb, ' .. + .. ~(Gaba + Ot/Ja + 01 ~ + ·.. 0.1) oJJs + ..• ·.. + .... Clab.)) + lJI(bt lfI(blll + bI ~ + ... ··· + b.'). b.I ). If weaeta - ";ba V blll + ba' + ... b.1l andfJ and ~ - z"; =,=V CIt Gil aI + ... +a..', + 0..', Ifw ...ta+bI .oo + +ba +oI+·oo '+

(CICIi o S:S (CXOa «'(Gal -- «'(Gil

the lut inequality becoJnel becomes thelut

:S 2(01 2(Clt1I + ·.. ... + .. I)(b1'al + ··· ... + ,I) oS a.1)(b bal) =F 2V Gal "'';-b-,I-+..-.-+-~-I (G,bt + ... aab.), 2"; GI' Gal + ... + Gal ";~b-=--1-:-+-'-'-' ~+-:~-=-I (Gtbt oo. + a.b.), or

z"; Gil CIt'+ ... +.. + ..1 ..; :It::V + ··· V b1a1' + ... + b.' b.1

(Gaba (1I1bl + ... ··· + .... a.b.)) :S I)(btl + ,··· .. + ,I). S (8-1 (Gil + ... ··· + .. tJal)(ba1 b.I),

or

V 81·+ Vblll + ... +b.'IGabJ+"· +b.IIGlba+ ... + +o.b.1 ..; 81+ ... .. · +..' + ..' ";bl .... 1 l :S .oo + .. I)(btll + ... S (G,I (0,11 + ·.. a.1)(b ··· + 6.1). b.I ), Gal + .,' + Gr.11 and ..; , . + ba If ..; V GI' ... +.. V ba b,1l + .' ·· b.l1 are both nOlllero nouero we can plOduct to I8t divide by their product let the cleBired desired inequality. If, H, on the other band, l Gal + ··· + a,.11 .- . 0 or ..; either ..; V 0,,1 V btl ba + ··, + bal -= == 0, then either 8, •••, .. b. desired inequality reducee reduces Ga -- ••• ••• -- a. a.. - 0 or bl bt -=- •• , - 0, and the cleBired o. to 0O:S SO.

+ ... + ..

+ ... + "

t

Par Clftll -II ,.., Corollary. For real

a., ".,

II~' AU"""" Ga, GI, Ga, •• •, a., Ga, ba, l,

ba, •• , ba bt, ,..• b. we have

..; (CIt + bJ)1 ba)1 + (Ga Co. + bs)1 .,' + (a. + ba)1 V(GI ba)1 + ... b.)1 :S ..; Gal + 01 + ·.. oo. + Gal -+-:~:-,-I .• S V G,I + lItl Gal + v'r-::~-=-I-:-+-:bI~+'---· V'-b-1'-+-".-I-+-.,-, ·-.-+-b-. thII write To prove t.bII

+

+ + ba)1 bs)1 + ··· .,. + (Ga (CIa· + b.)' b.)1 01 + ... 0.1) + 2(aa6a + Gt1 ·.. + +.1) 2(tllbt + a.bt Gtbt + ... ··· + .... a.b.>) + , . + 11.1). + (b (bal + bI lJ.1 + .··· b.

(Ga + btl bs)'l + (CIt (ae (at (8-1 - (Gal

l

l

I

).

36

III. METRIO MIlTRIO SPA-OilS 8PAOl!l8 lIt.

Schwarz inequality the Jut laat exprestrion expreeeion is 1888 By the Schwan lese than (\\' ot equal to

··· + a,.1) + + ... +~IJ,.::...:I):..-.:---c-_---,---= + 2V-O-11-+-Gt-' + -.b,' + ... · ·· + b,,1 + 2V til' + CIt' + .... •-+-a,.-I + IJ,.' v' V bl bt'1 + bi b..' + (bl' (bl' + btl bi + ... + b.1) b..,)

(G,I + ~ (til' CIt' +

which equals equale

(v' bl 'l + bi bl l + b.I)I. ( V GIl til' + 0/CIt' + .. ...· + 0. IJ,.'1 + v' V ba + ... ••• + b ..·)'. Thus (/It (Ot

ba)' + (tit ba)' + ... (IJ,. + b,.)' + bt)1 (fit + bt)1 ··· + (a. b.)1 S (v' btl + lJ,1 ··· + bat)l. (V Gil til' + fJt,1 CIt' + ... + Gal IJ,.' + V ba' bi + ... b..')·.

The desired result reeult cornel comes from this lut inequality and the comment that ifOSsS,thenv'i Vi, ilO~.~"thenVi S Vi. We can now verify the triangle triaDlle inequality for Ea. ~,,), J!lra. Let pP - (~l, (Zit ••• , z..), (fIl, •. •••· t'.)' t 11..), r .. ). Then q - (rl, 1m ('1, ... , ... Ita). d(p, r)

..V (211 - ,tl' (z.. - ... )' -V(:1:1 .,)1 + .··. .· + (~ -..)1 .. V«%I-1/1) V«SI- fls) + Ull. <111fI..) + (,. (fl.. - 1,.»1 ...», .. - ,S», all)1 + ... + «z.. «~" ~ fI,,) s:S V (2:1 (211 - fll)1 fit)' + ... + (z.. , .. )' + V (fit ,s), + ... + (U ...)' ··· (~ 1/,,)1 (1/1 's)t ··· <71... - ",)1

by the CoroUary, 10 that

d(p, r) r} tl(P,

S ~ d(p, tl(P, q) 9.) + d(q, r).

B- is Thus J!lra it a metric space. ep&Ce.

We note that J!lra generalization of the first example, since Jll ga il is Eta is a generalilation == I" Ip - II. ql. simply R with d(p, q) :II (8) B is a metric &paCe E 1 is sublet of B then 8HI1 can be made (3) If 1£ apace and 1£1 it a subset space in an obvioUi diltance between two pointe a metric apace obvious way: the diatance pointa of E Bl1 is it the .me .. as the distance diatance between them when they are considered pointl pointa of B. That Bit Bl , topther together with its ita metric, aatiafies it immedi_tides our four axioms axiOIDl is ate. E l , with ita is called a n&bIpace its metric, it ~ of E. B. Note that by takinlsublete taking aubaeta of Euclidean space ,pace we set get an infinite Dumber number of metric .pac., Space8, in a tnttre01 mendous variety of Ii. ail8ll shapes. mendOUI . and .hapee. q E H, define d(P. d(p, q) - 0 if (4) Let E be an arbitrary let and, for p, fEB, p - q, d(p, 9.) q) .. - 1 if p pi q. This iI it clearly a metric apace. ep&Ce. It is a very P fl, tl(p, jf p" hut special kind of metric space, quite unlike the previous examples, 00' illuetrates illustrates nicely the generality of the concept with which we are dealing.

12.

0 . . . AIID AND CLOUD CL08SD 111ft 811ft

11 IT

In this text we shall for the moat most part be interested in g. E- and ita aubspaces, .but 8paceI will allO appear and inde.l indeed will ,but other important metric IIpaceI IOmetim. be introduced to prove thinp itlelf. Wop about .II-. itaelf.

0/ meIric . 01 Ute u.. ""'"' .. . H, E, . ,. d(P.. Pta) p.) S tl(Pt, d(Pl, Ptl d(p., pal d(Pl, PI> + d(ps, Pa) + ... ··· + d
"ropoalelon. 1/ PI, PI, .•.. Prop08'elon. 11 Pt, · · , p. en are poW.

This cornel trialll1e inequality: com. from repeated application of the trianPe d(Pt, p,,) d(PI, p.)

S d(Pt, P.) S d(pa, d(p., Ptl PI> + d(pa, d Pa) + d(p., tl<Pa, p.) S ... d(p,., Pt) PI) + tl(p., d(p., p.) d(PI, Pt) PI) + d(p., S d(Pt, d{1Jt, PI> Pa) + ... ··· + d
Propodtion. "ropoalelon.

11 p, f, I, r are poi" o/lAe . H, B, IIaen lAm 1/ poi"" of 1M metric . .,.. Id(p, r) - d(f, S d(p, 4(9, r) I s tl{P, f). I)·

This i. is esaentially that, 8III8Iltially the well-known fact of elementary geometry that the difference gf Bidee of a• triancle trianl1e is illell than the third aide. side. To prove of two aides I. . ~

it note that d(p, r) S tl(P, d(p, q) f)

+ d(q, d(f, r)

and d(f, d(q, r) S de"~ d(q, p) + d(p, r),

which can be rewritten d(p, r) - d(f, r) S d(p, ,) d(P, q) and d(f, r) - d(p, r) S d(f, d(" p),

which combine to

Id(p, r)

- tl(f. d(f, r) I S S d(p, tl(p, f). I).

12. '2. OPEN AND CLOSED SETS. DeJinieioru. Let E be •a metric apace, DeJinidom. space, PI E B, H, and r > 0 a ." realnUlllber. awJ aumber. opm ball in E 01 0/ ceraUr center ,. PI _ ,adiua radiUl r, is Then the open i. the aubeet IUbeet of B pftll liye by b7 (p E H B : dc,., d(JJt, p)

< rl.

The cloMl ball in E 01 ceraUr PI ,.,." aM ""'"" radiu r, is II 0/ center (p e E B : dc,., Ip tJ(JJt, p) Sri. SrI.

II can. be no miauDdentaDdIDI milundent.uadiDI about If there eaD abou' what wW the me&rio...,. me&rio ~ ... B II, ODe often epeakl apeak' of the "open (or clOlld) oIoIed) ball of OID_ " ucI1'1diU one and tIMIiU r". ~'. epeakI of an "open (or oIoIed) oIoHd) ball" When one apeakI balItt one ODe II\I&Da II\IIM .... an os-a opa ('" (. clOIed) baD of lOme center" . . . . and 1OIDe. IIOIlle ....u. radiuI ,, > 0. e1011d) center Po in the metric .... Bya "baD" is meant an open' opeD' ball or . . . . W. bIIJ. If By a "baD" a .cIoIed U our ............ metria ..... B fa ia precedinl termiDOlolY· terrninololrit ordinary 3-epaee B' then the precediDI iI in ill MCOId aceord with..,..,... with .,.,. Jancuace: an open ball baD in ~ B' is it the iDaide inaide of IIOID8 ephere while. ca.I day laDlU&le: oIlO1D8 cIoIed

oeD'" "

m......o ...... m ......o .. ~

•

ball It iDIlde 01.,.. on the ...... WI ie the iDIide 01 eome ..... t.opther wRb with the point. OD ......, the aDd IIdii rIdii 01 the baDe eeaten aod aDd radii . . . . aod baUa .,.. beiDc the eeoten IIdii reepectinl7 reepectivel7 01 the die . ...... . In the plane pIaae .. AD open opeD bell ball •ie the iDIicIe . JII aD m.icle of lOme circle eircle while whU. a& .... ball. the iDIide 01 a& eirele ciNle topther with witll the point. on on the circle. cln1.. In ..... ball ie the . . . meuio meUio . . . . baUa baIIa ...., uball" (of. otha' mat look eve ... 1_ like an aD ordinary "be11" J'Ia. 7).

.....,. PI 7. A .................. flI.pIaDe. 01 the piau •

aiwa ... .... b:r • -I6I.,)e.:->o.,>ol. - 1<-' ,) e .:- > 0. ,> 01·

w.

if.,. e a, _<', <., . . .. . .,..........., ..........

We nc&1ltha, nca1l ..., if ." . . _...,.ltthe_ "a.'lIthe_

tIae ~ ................

a

e. :•

(e,') (- e a : a < • < 'I' '1 • (a, ') - 'a while the.."........, the .... ........,. ..... the_ wbDe ........ a"'" II the.-

11""._.....

r-.'l-Iae.:a~a~'l .. l-.ll(-e.:-~-~&).·

-<-<& a
or

the Iaequalllitl iDequalltiti __ ~ < +& <&_ o+b a- a+'
_-b

o+b b-o -2-<---2-<-2or

...

.

a+'I<"-a Ia-,--,-,

'.

......................... ....... 1ImIhed ..... TIle aymbol (...) ........... . . . . ........ . . . _ . . . . . . . pUr vi ...... iii R (_, wbich.dIe . . . Wq, . , . . " ..,. Tbe . . . . to be .... to (G, ') IIIauId alwaJII be . . fl'Olll . . COD. .

12.

OPO CLOUD 8II'1'II 8ft8 0 .... AND 0L08IID

19 It

80 10 that that. (a, (a. b) is the open opeD. ball in the metrio meVio space apace R with center (a + b)/2 and radius la, b) b] ill closed ball with wiUl the aame I&IDe cen_ centel radiUl (6 - 0)/2. a)/2. Similarly [a. is the cloeed and radius. U PI E closed) ball in B R radiua. II E R and r E E H, R, ,r > 0, then the open (or cloaed) with center PI and radius r]), 10 10 that radiua r is
"'1----

.. , _ - - - rr - -...,....- - - r, - -.... --_'~"_------tal II

II,

III

,1

p.-r p. - r

iD.

PI 'P.

II

I 1

p.+r 'P. +r

l'Iouu with aeatel' ceater,. radlUi r. l'Iova 8. Ball iD • wlda PI and radiUl

DeJinltlon,. B is ia opeA open if. if, for each p E E 8, n.Jlraidora. A lubset aubaet 8 of a metric apace E 8 contains contaiua lOme open ball of center p. Int.uitively. a aubaet. Intuitively, lubset 8 of the metric apace lpace B E is open if 8 containa contains aU aufficienUy near any liven point of 8, but t.hia this all pointe of E that are aufficiently propen)' ltated repeatilll the definition Juat JUit liven. property can be . .ted preoiaely only by repeatiOl Note that it iI open or not unleal unlell 8B is a it. makee makea no IeDle IIeDIe to .y I&y that a let aet 8B is _bset of lOme apecifio metria apace B. IUbiet apecific metric

Propodcfora. For 'or .. , metric PropoRtion. -11 traetrie Ipace apace B, .. ... (1) ' " _ _ /IJ flJ if open (I) "' _ _ B iI .. . .. (') 1M"" open (3) '" tAB union tmion 0/ .. an1/ aubaet. 0/ oj B .. ia OJNR open , coll«:tion collIJction 0/ open OJNR ...,.,. itaWlICCion 01 0/ a ftAtu AVtRlHw 0/ (4) the iftMr.."iaA ft",m ",umber 01 OJNR open ...,.,. lUll"" 0/ oj E E ia iB opeA. open.

1M""

fint item is trivial, t.bouah The proof of the firat tboUlh perbapa perhaps trioky tricky for beginuen: we have bave to show ahow that "for /IJ there is an open ball wch ufor any p E E~ ballluch begiDDen: t.hat. ...", • . •... a etatement atatement that is automatically true ainee that lince there is AD AO P wch such that p E flJ. The aecond I8COnd item is equally trivial; indeed any ball in B E is contain8d in B. Item (3) is also al80 clear. To prove (4), let 8., contained 8 1, ..•• ••• J 8. be open aubaete of E f"\ 8 •. For ii..... lubseta B and p E 8. 8 1 (f"\\ •.. • • • "S•. II:: 1, ... , A. ft, each 8, Bi is open, 10 80 exiate a real number r, re > 0 auch there exists luch that the open ball of center p and ndiua r, re is ia entirely contained in 8" 'p and radiua S.. Then the open ball of center center·p radiua min minlr., ••. , r.1 ia radiUl Irl, ... is contained in each 88,e and is contained therefore f"\ ·... ThUl 8." in 8. (\ .. "8 n 8..•. Thus 8 1 ('\ ... • • • "8. ( \ 8. ia open. eo far occura The word "open" 80 occurs in two contexte: contexta: we have open balla leta. That no errors errore can arise ariae is a conaequenee and open Beta. conaequence of the followina followin. ~~ . result.

40

IU. In. Mm'RIO M1I'I'IUO SPACH 8PA.CH

Propoeition. 1n Gntl metric apace, _ open bt.all ta _an open •• Proposition. In O"Y rpace, (1ft, lHJll it Nt. For let E be a metric space and eoD8ider opeD ball 8 of center ". comrider the open and radius r. We have to show that if p E 8, then BOme lOme open ball of center p is ill entirely contained in 8. Figure 9 gives the idea of the proof: if "E pE 8 then the open ball of center" tl 0 and there actually exiItI p and radius,. If,f is thillatter ball then d(p, f) I) < <,r -- d("., d(Po, p) radius r - d{JJe, d(po, p), pl. If ill in thilllatter and therefore d(p., q) d(Po, p) + d(P, q) < d(po, d
+

+

tJ(po,,) - " , , # - o ", r - "(P •• ,,)_~.(/

d("",)

Plovo ill all FJouu 9. All An opeD ball iI aD opeD open lit.

A. •a colllequence propositions we can ueert &llert that the collleqUenC8 of the lut two pt'Opoaitiona subeeta of B E are precisely preci8e1y the unions uniona of open balls open subeets balla of E, that ill, is, any subeet is ill such a union fm fact it is ill the union of aU epen sublet all the open bal. balls it contains) and any such union ill is an open set. Bet. The difference between (3) and (4) in the fi.I'It first propoeition is iI to be noted serioualy. seriously. Item (4) is ill no longer true if the word "finite" DOted "finite" ill dropped. CODIider in r the open ball, For example, colllider balll with center the origin orip U, 34, ... ; the intenection (0,0, ... ,,0) 0) and radii 1, ~, ~J intenectioD of theee these open bal. is ill just the origin itself, itaelf, a eet let which ill balle is cleariy clearly not open.

Dtr/in.tfon. ite complement Dft/I.... tfo". A subeet IUbeet 8 of a metric apace space E il ie clotetl eloNtl il if itl eB (that ie, ii, all pointe in E which are not in 8) iI e8 is open.

Ploua 10. ()peD FJoUBIIIO. Open Uld and cIoIed c10eed llta eeta in P. BI.

12.

0. 0 .... . 4JfD AND ClLODD eLOIIID . . .. .

.1 41

ahow that a closed cloeed bell ball is a ekIIed clClIed As before, to avoid trouble we should show set. Here are the statement and proof, quite anaIopue analOlOUB to what W8I was done above. Figure 11 illustrates illUlltrateB the argument. ar;ument. Propoaltfoft, I,. Oftll mdric .,ace, G cloaed ball it G . . . . , ....

e es.

Let 8 be the closed ball of center Pt Eiland PI E B and ndiUl radius r aDd 1. let ,p E e8. OpeD ball Then d(,., d("., p) > r, 10 eo d(p." d("." p) - r > 0 and we can CODIider the opea opeD ball of center p and radiUII radius d("., d(Pt, p) - r. For any point f in the latter 0J*l q) < d
+ d(q, p) -

>,.

d(p., d > r. Thus ThUll the open ball of center p and radiuI radius d
d(p, f)

~

lIcp-,p) i(Jle,,,) - rr

a'"

FJGtlUI 11. n. A A c10Ied cto.d WI ball II ill a cloled lilt. FlG1JU

Analogous to the first proposition on open . . we have the followiDc leta loI1owiDc result. .

Oft"

Propoatfoft. For Oftll meIrtC metric Ipoce H, Propoeldo... Por apace II, Ute aubld B E .. ia cloI«I (1) 1M (I) 1M Ute IUbaet Il1 it ia ..., cloaed (') au'*' 12J (I) interMditm 01 0/ emil coZl«:UorI 0/..., ...... 0/ 11 it . . . (8) 1M Ute i"""aecMn coll«:tiorI 0/"'" 0111 (of) 1M Ute Uftion ft,f.- "",..".,. oJ ..., ...",. 0/ oJ 11 Bit ...... (.f) tlftiota 01 0/ •G ~ AU"'" 0/..., ...

""'Nt

Oft"

TbiI reeult NIUlt followa fo1lcnn immediaW7 bnmediately from the analopat PJOPOIitioD propoIitioD lor for Tbie open eubaetB. eubeete. For B and 12J Il1 are the oomplemen. complement. of RJ f4 and aDd II 11 nepeetheIJ, reapeotiYeIy, which are open, 10 eo that 11 B and RJ Il1 are ..... cloIed. Parle P..... (3) and aDd (.) (4) ,.... ,...... fl'Olll lfOID UDion, and . ...ftIIa ... the fact that the eomplemeot complement of an intenectlon IntenJection is •a UDioIl, aDd . (bottom :EuitoIa., EuitcI8., pap 8). For (3), we note that the complemeDt compl. .t of tile the intenection cloIed au'" of 11 E is the uDioD union of the inWleetion of any collection of cloeed

fI

m.........C . .ACD

complements compIementa of theee these closed clOled subeeta, au....., that is the union or of certain open leta, which is open. For (4), note that the complement of the union of a finite cloeed aubeets subeeta is the intenection interaection of the complements of tbia this number of closed clOled sets, seta, that is the intenection intersection of a finite number of finite number of closed leta, which is open. open ..ts,

II p, ..... t are dlatinct dlatinot points point. 01 If of the metric apace B then there are balla (elther open or clOHCl) contalnlnlone point but not the other, for example (Iilther clOlClCl) oontalnlnlone ciOled) centered at p or q and of radius I... I. . than any ball (either open or closed) cI{p, q). Thus Tbus any point is the intellection intenection of 01 all the closed balla ball. containing containinl d(" is a closed clOled set. By (4), onr any finite it, provilll (by (3) above) that a point iI oJ a Mric tndric qoce IJHIC' if iI cloa«I. .__ . . . . 010 The oomplement complement of an open ball is a closed set, as &8 is i8 the closed ball of again by (3) the intersection intel'8eCtion of these two the .me center and radius, 80 &pin seta is closed. clOled. Thus for any ,. PI E E "8phere of center Po leta B and any r > 0, the "sphere radius r", is, the set (s E E : d(p., ... rl, r}, is closed. and radiUl r U , that ii, d
as-

[o,b)(0, b) - (zER:aSs
l(sl, ••• ... ,,:c.) Sl> ot I(sa, ..) E EB-:: Sa > al ill open. For let p - (Sa, (:ct,:ct, ZI, ••• .•• ,,:c.) . .) be in this aubeet lubset and consider the open SI - a. O. If II q - (YI, ..• ,,Y .. ) is in the latter ball of center p and radius Sa
(Sa - ra)1 S vi V(:l:l Yu' + ... + (.. (:c,. -

r.)1 0= d(p, q) < Sa - a, tI .. )'''' <:I:I

80 'a til - Sa :l:a - (Sa (:l:a - 'a) Ya) ~ Sa :l:a -Isa -lSI - raJ> YII > Sa :1:1 - (Sa (XI - a) 10 (1) ... a, ca, 80 that the is in the set t(:l:a, Eli:": :1:1 > > a}. point fq ill (Sa, .•• ,,:c,.) ..) E E" : Sa cal. Hence this latter set is •• •,,:c,.) :1:1 < < cal a} is i. open, and for any let t(:l:a, (Sa, ••• ..) E EE" : Sa open. Similarly the set

... , ", '" the leta i• - 1,2, 1, 2, .•.

«ZI, Zf> 01 and «Sa, «Za, ... ... , S.) z ..) E Eli:": Zf < 01 al «Sa, •. •••. ,,:c,.) ..) E EB" : Sf> E" : Sf .... are open. Consequently Colllequently their oomplements complement. «:ct, ••• ,s,.) , z..) E.8" Eli:":: Sf s. Sal S CIt and «Sa,

areao.ad. aN"'.

((Zl, ••• , S.) s ..) E EE" : Sf Zf ~ cal a} (Sa,

........... .

.,st. .............

u ......................... ea e ..... < ..... aad 4.
. of pointe polafll then dae the . let

-

« •., ,..,-.) e .. «~., B- :: ~ <., < ~ < II. for each MOh i-I, .. ..., til tal 0 .. ,

..,

0,

ia ca1led an &0. . . . ..,.., aad the . . of pointl poin. ..",. ita II- and let

1< • ., ...", ..) -.> e" ell-:: ~:s; .. :s; .. I<~ 1'4:S; II. for eaoh each i-I, .. ...0,, til tal

"0

..uaa

II ca1led cIot«l Ieneralilel the UIUAl notion of open ia ca1lecl a . . . ..",. ..,.,.,., ita B", (Tbia (Thia poeraIlHa opea Atl .OJN" . . (or doted) i,.",.,., in and cloeed interval in R - B1,) Ata cloted) italmlol ita BE- ia '- CIA em open OJNA (or"', ,...,."...,.,,,) auheL _ _ For IUOh latenectioD of the 2ft (or c1Metl, ....,..,.,,,) nch a let it ia the interlectiOll 2ta IUbeete PVeD IUbaete pven by the aeparate eoaditiODl conditiooa 8, 4, < .1, s., . ,., .. < .. Sa,, ". ba > .1, ~" .. ,, b. >.. • ., ,. " CIa S :S; .., .., ". ADd theee > Sa (or 8,:s; 4, S~., ba ~ •~"., •• " b. ~ ..) and theIe latter 1UtJeet. aU open opeD (or eloeed), cloeed), hence heace 10 latenection. IUbeets are all 80 is their interl8ction. ,810)

0 0 0, •

0 0 0 ,

•

0 0 0, . .

0 0 0,

. .)

... ---.' -'-I

.. --

.. ---I

..-,

,

~-

.,.. I

l'Iauu 11. A. oIoIed obecIlDt.erYal1D l'Jouu 12. iDtervaI iD ...

lhtIbdefoA. ~efo".

A IUbeet IUbM S bou,."., if it it 8 of a metric apace E is ia boutadecI is contaiaecl contained

in lOme ball. In this definition the ballla ball in question may be either open or cloeed, eloeed, for any open ball it contained in a cloeed baH ia eontained ball (for example the cloeed ball of the same eloeed ball is contained in an IAIDe cenw center and aad radiua) and any clOl8Cl &0. open lAme center and &0.1 any l&l'ler ball (for example the open ball of the eame Jarpi' radiua). ~ bounded let, consider eloeed intervale in Aa an &0. example of a boUDded coD8ider the open and cloeed .. cliaoUlled above; tbeIe aU boUDded B" diIoUlled tbeae are all bounded IIinoe linoe the let

«.., .. 0, ..) e" ell- :: ~ S .. -1, "0, «~.....,..) 1'4 :s; S II. for each ii-I, .. " til tal ~:s;

, CIa) and radiua of magniis contained contaiaecl in the cloeed ball of center (a., 0... . 0,.) ADd radius tude V(b, Gl)1 CIa)', V' (fI, - -.)1 (b. - c.)'o

+ .,. + 0 0 0

"44

m.

MIlTIIJO IPA01I8 IIPA01l8 MIlTItJO

If 8 is a bounded subset of the metric space E then 8 is contained in Po, where Po is «my some ball (either open or closed) with center Po. CJft1/ point of E. IIOme ball, say the closed ball of center Pt PI and For since 8 is contained in some radiU8 ball of center Po and radius radiua r, then 8 is also contained in the closed hall radiue .. ,r + d
.

Prop03Ltfon. .A A ftOftempt1/ nonmtpty doted cloaed aubNt BUb_ of R, if it it ia bounded from above, Propoll'tfon. Ma a ".eateat element and if it ia from belotD below Ma a~ leaat element. Aa G"""'" elemmt GfId it bounded botmdetllrom Aa G elemmt. Let 8 be a nonempty subset of R and suppose, for definiteneee, definitenell, that 8 ie bounded from above (the proof being almost the same if 8 is bounded is I.u.h. 8. If G a fi 8, then tJa E e8 and since e8 is i8 open from below). Let Ga -= - l.u.b. exiets a number E > 0 such that the open ball in R of center tJa and there exists ie greater than radius E is contained in e8. This means that no element of 8 is a-E. i8 an upper bound for 8, contrary to the 888Umption tJ - f. Therefore, a tJ -- E is 888UJDption atJ --l.u.b. l.u.b. 8. We conclude that atJ E 8, as was to be shown.

e

f S. CONVERGENT SEQUENCES. 13.

Let Pt, Pt, Pa, PI, ••• ... be a sequence of poiute points in the metric space E. It may happen that as we go out in the sequence the points of the sequence "get arbitrarily close to" some IIOme point p of E. This is i8 illU8trated illuatrated in Figure 13, terms of the sequence. at first oecillate oscillate irregularly, then where the various tenns proceed to get closer and closer to p, in fact "gang up on" p, or "get arbigive some trarily close to" p. The purpose of the following definition is to live Bellle to the intuitive woJds words "get arbitrarily cloee eloee to". precise sense

. .

. o

"1

It.

o

P, PI

p,

. o

PI

• PI P. o

p.

0

't. 0, o

. o

'I".

,

,.. 1• FJoU1Ul FJoUlUl 13. A A eemverpnt eODvergent eequenee of points pointe in 8 8'.

,3.

COllvmlOaNT BIIQUIINCIIII

...

Definition. PI, Pt, Pa, Dejinition. Let Ph PI, . .. be a sequence of pointe in the metric E E is called a limit of the sequence eequence Pl, PI, PI, Pt, PI, Pa, ••. ••• if, space E. A point l' pEE liven any real number e > 0, there i. poeitive integer N such that is a positive d(p, ,,,) If the sequence lequence PI, Pt, PI, Pt, PI, Pa, ••• baa a limit., limit, p,,) < • whenever nfa > N. II lequence we I&)' .y we call the sequence convergent, conHrgent, and if ,p iI is a limit of the sequence that the tAe .quenu aequenu contJerfe. 10 to p.

COfUJ"".

Let us make a few observations coneeminl concerning this definition. The main one il poeitive integer N having a certain is that given liven any e > 0 there exilts existe a positive property involving e. Thus Thu8 N uaually usually depends on e, and it. it would have been more precise to write N(e) instead of N. However the extra precision obtained by writing N(e) in.te&d instead of N I't'Jlulte results in unnec888&l'y unneee.ary not.&ticmal notational juat tacitly und8l'lltand understand that N depends on e confusion, 80 from now on we just and stick to our shorter notation. Note that we do not even care wbat. what specific N goes with each e > 0, the important thing being that for each fe > 0 there exists,ome lOme e > 0 we existe 30mB N with the desired property; if for lOIIle have a certain N with the desired property, then any larpr IUler N would do equally well for our given e. Thus in deciding whether or not a& sequence eequence of liven •. points is convergent, only the terms far out count; that is, ie, if we obtain a tenna of our original sequence, eequence, the new sequence by lopping off the fint fil'llt few terms two eequencee oonvergent (with the _me I&me limits) limite) sequences we have are either both convergent converpnt. or both not convergent. iI that Another observation on the definition of limit and convergence it some specific metric space these concepts are always relative to lOme lpace E. ThUl Thus liven sequence of pointe in B, it might happen that for the given H, Pt, PI, Pt, PI, Pa, •.. ••. , the condition in the definition of limit holds holda for a certain point p of B', E', i. some metric apace where E' il Ipace of which B it is a aubepace; subspace; the sequence eequence would could not call it convergent in B unleaa then be convergent in E', but we eould unleea p E B. Thus in using the notion of convergence we knew that pEE. converpnce a& epecific specific metric space must be borne bome in mind. A. an easy example, the sequence 3,3.1,3.14,3.141,3.1415,3.14159, 3, 3.1, 3.14, 3.141, 3.1415, 3.14159, •.. . .. converges in R (to the limit 1f), 'If), 11f} of R. but not in the subspace e efT} speaking' informally of a convergent sequence of points In speaking'informally pointe Pl, PI, PI, Pt, PI, •.. ... uthe points in a metric space E, one often says that "the pointe of the sequence set let clOier and closer cloeer to the limit", but tbi. thie need not be literally tNe. For closer u we move alonl along in the sequence example, .. lequence the pointe poin" of the sequence may at fint fil'llt let set nearer and nearer to the limit, then move somewhat IOmewhat away, then get nearer and nearer &lain, again, then move eomewbat let IOmewhat away, etc. ete. Thua while terms of the flequence sequence of pointe points in R the term.

)i, J.(, U, 1, ~, }i,.34,

...

do get closer and closer r.1oser at each .tage stage to the limit sero, zero, this is not litenlly literally true of the convergent sequence

I, ~, ~,U,~, J1I, ~, )i,~, M, 3rf,~, 1,

.... ••.•

m.lI ......c ....... m. IIJmUC .PA'"

46

The pointa points of &• eonverpnt a1IO happen to be all equaU, equallfl near 'l1le converpnt eequence IeqU811C8 may &leo sequence the limit, since for any pEE, the lequence

p,p,p, ... p,p", coaverpa to p. CORV_top.

ProJHNI t Ion. A MfU"'C' Pl, PIt PI, Pa, •••• •• oJ 0/ pH"" poi"" in PrGIJOfJtlon. ....... "" ., ,...., .... at ".., OM limiL limit

CI G

meCric .,.,. ".,. •II Acaa mtCric AaI

For IUppoee , E E are both limita limite of the eequence. sequence. For any IUppoll8 that p, fEB there are positive inteprl integera N, N' H' auch such that d(p, Pa) < • if "n > N pa) < e • if ntI > N'. N ' . If we chooee an integer n > max IN, {N, N'l we and d(f, d(" Pa) must have Pa) < .. ., d(I, ,.) < ., mUlt bave d(p, p.) d(f, p.) e, 10 that

•e > 0

Pa) + d(p., ,) q) < e • + ... d(p, ,) f) S d(p, p.) • - 2•. 2.. If "(P, tl(p, f) 0 we pt. _ & contradiction by chooeing ,) ,. "0 choosing •E S d(p, ,)/2. q)/2. Therefore we mUlt have "(P, tl(p, ,) f) - 0, 10 tW tbat pP - f.

Tbua lequeDOfl hu makelMD18 Thua &a converpnt lequence baa &a unique limit, and it mak• •DIe to Ipeak of 1M tAl limit of &• converpnt oonv_~ leQuence. lequence which ie is not conlequence. A lequeBCe v.pot verpnt of COWIe coone hu baa no flO limit. 'l1le ltatement tbat pointa Pat (in & metric The .tatement tbM the IeqUeIlC8 aequence of pointl Pl, "., PI, PI, Pa, .•• • •• (in. Ipue II written CODCllely concilely .. I ' " B) converpa conveIIeB to the point poio' p (&110 (a1IO in B) ia u

-....

lim". - ,. p.

For & ftODCOnverpnt Pa, PI, PI, . •• of poln. point. in &• metric noneonverpnt IIqUeDOe MqUen. Pa,", IpMe p. .. meaninl1-. In this we &110 .,. that apace B the exp...wn exp..-iOD 11m lim ,. ia --.n1nII-. tbil cue .e a1IO . , tba'

-.... .............

-....

lim p. "dOlI "do. not exiIt". exiIt"•

If Ga, (01 any kind of objecta) object8) and if "1, na. tit, Gl, a., CIs, CII, Gi, ••• ie is •& lequeDOe eequenoe (of tie, ••• it & a 8trictly inereuiDl leQuence positive intepn (tbM is, lie, atrictly increuina lequence of poIitive intege... (that ie, flat "" 1&1, ft., ... are positive poeitive inteprl intelen and na til < na At < lit til < ... ...)) &hen then the die "" na, .•. lequence "-t, ..... a"., ••• is Gl, CIt, IeClU8llC8 ie called a& ~ of the eequenee leClueDCe Ga,",

,

a.., a.., a.., ...

..........clon. A"U ..........elon. An,

~

01 G co,.".,.,.,., convu,mt oJ

aequence

oJ

0/

poif'&tI poiftII in

G

tndric IptJa c:tmM'fa eonHrfIA 10 1M eM .... limit. tnIfric IJNIC4I

.-....

p. - p and let "I, na. tit, & strictly increuing increuinl For let lim lim,. At, na, fta, .• .. •• be a lIquence intepm. Given any e• > 0 there is a positive intepr integer N IeClU8llC8 of poeitive positive inteprl. IUCh that t.hat c1(p,,,.) II(p, Pa) < <: •e whenever n > N. Since "- t! ~ m for all positive lUCIa In ..... WI, .e we have m > N. ThiI TbII meanatlf'. meana -*,t. lD.... bave c1(p, tI(p, p..) < • whenever '"

-...

lim,.. lhow. [Note (Note that m in the 1Ut lut lim ,.. - ., p, wblob it II wbat we wanted to sbow.

-....

f'-mula. GImuIa • Juat & "dWD1D7 ...wu p..] jUit. "dU11UD7 variable", . .u "n in the expNMion exPrelliOD lim Hm 'A']

18.

OONyp·PRQVU" OONYDGDT 8QVUc.

.., .1

The Precedinl PrecediDl result reault lenerali.. pnerali. . the fact that if a finite number of tenus convergent eequence, sequence, the ..... resultiDl tenns are lopped off the beginnilll beginning of a COIlv8l'lent ting sequence converges conv8l'les to the same aame limit. Call a sequence points PI, PI, PIt Pt, • •• in a metric apace space botmdefl bouftded if eequence of pointe the set of points Pa, ... ••. 1 ia bounded. It it is euy easy to ahow .how that any pointe (PI, PI, PI, J Ie oonveJ'lent I8queuee convel'l811t sequence 88qU8llCe of pointe in a metrio metric apace 11 Ie bounded: If the lIq\lellce PI, Pt, .. oonv. ._ to the point P, p, plok piok any • > 0 and then find a PI, PI, ·.•• converpl positive d(p, p.) <. whenever ,,> N. Then poeitive integer N luch that tl(p, (PI, Pt, Pa, PI, •... ie contained in the closed ball of center P {Pl' • • I, is p and radiue radius max (I, (e, d(p, p,)., d{p, p,), ... , d(p, PN) )• tl(p, P.)', tl(P, PI>, tl(P, PH)J.

We recall that a closed lubset subset of a metric space was defined as &8 the complement of an open subset. subsets lubset. Thus Thue the knowledge of all the closed subaete of a metric space Ie is equivalent all the open subsets. It equiValent to the knowledge of aU is all the open subsets of a metric space ie also true that the knowledge of aU determines which sequences convergent, and to which limits. eeqU811Cell of pointe are converg811t, For it is Pt, PI, pa, ... ie immediate from the definitions that the sequence PI, PI, conveJ'1e8 to the limitlimit' p P if and only if, for any open set U that contains the converges point p, there exists integer fa > N exiata a positive integer N such that for any integer" we have p. E U. The next result will tell ue us that knowledge of all the convergent vel'lent sequences sequencea of points in the metric space, together with their limits, determines the closed subaeta subsets of the metric lpace. apace. Thue Thus any ltatement 8tatement concerning the open IUbeets IUbsets of a metric apace can be tranel&ted tranalated into a ltatement concerning subeets, which itaelf itself can be tr&nsl&ted traD8lated into concerniDl the closed cloeed subeeta, ltatement ooncerninl concernin& converpnt another atatement convergent eeqU8llCeII aequenc. of pointe and their Thua there are three languages capable of making limite. Thus limits. makinl eIII8lltially 8I88Dtially the ltatemente: the language languap of open sets, that of closed sets, eame statements: seta, and that of sequencea. As AJJ one would expect, however, limple convergent sequences. simple statemente statements in one language may translate into complicated statemente statements in another. We eha11 therefore \lie UI8 all three languages simultaneously, elmultaneouely, or rather a common shall (88 opposed language including all three, always striving for simplicity (as expreaaion. to purity I) of expre88ion.

""'Bub"' Nt

La 8S be Ga oJ 1M Theorem. Let oj tM metric IpGCe Bpace E. T"'" Then 8 if is cloNtl clo,«l if aM and only iI, i/, whenaJer ~ pl, PI, Pt, PI, PI, Pa, ... · .. if i8 a uquenee sequence 01 poi"" poinlB oj 8 t4at that if ill COfWergen.t convergent mE, tIHJ 1uwe in E, we luwe lim p.E8. p. E S.

-

• -.ell)

There are two parts to the proof. First suppose that 8S is closed and that Pl, Plo Pt, Pa, PI, .• •• is ie a sequence of points of 8 that convel'lee converges to a point p muat 8how show that P E 8. If this thie is not 10, E e8. Since of E. We must pES. 80, we have p pEeS. e8 is Ie open, OP811, there Ie such that e8 eS is lOme eI > 0 8uch eB contains the entire open ball radiua I. Thua if N ie of center p and radius e. Thus is &a poeitive positive integer such luch that tl(p, P.) p.) < e whenever" > N, we have p. E e8 whenever" d(P, whenever n > N, H, &a contraprovea the "only if" part of the theorem. diction. Thie aWl ahoWi that p E 8 and prov.

48

Dt. MISTRIC BPACB8 m.MmTlUCIIPACilII

To prove the cCif" "if" part, suppose 8 C E is not cloeed. closed. Then e8 iI is not open, and there exists a poi~t luch that any open ball of center 'P p point p 'P E e8 such contains pointe P... E 8 points of 8. Hence for each positive integer ft n we can ean choose 'P d(p, 'P ..) < l/n. Then lim 'P p. E 8 and 'P C! 88.. such that d(P, P.) p".. .. 'P, p, with each '" p E!

-....

This shoWl that if the hypothesis on convergent sequencee sequences holds, then 8 must be clOled, uif" part, and hence of the closed, completing the proof of the "if" whole theorem. The metric space R has special properties that are not shared lhared by all metric spaces: its elements can be added and multiplied, and they are ordered. For want of a better place, we insert here the relevant properties of sequences of real numbers. Proposition. PropotJidon. 1/ Ot, aI, Ot, CIt, Ba, /Ja, • •• OM and bl, ba, bba,a, ••• orB o.re cmwergetd convergent aequmce. aequence. t , bt, 0/ rfUll rBCJlnumber8, a and tmd b rapedively, respectively, Uatm oj numbers, tlJ'WI. limits tJ Uaera

un''''

-....

+

+

(a" + btl) b,,) .... lim (0. == fJa + b

.....

l im (a.. (a" - b ..) == = atJ -- b lim b.)

-

.... lib

lim a"b,.

. , in CGI8 CfJ86 b aM tmd each eoc1a bIt b.. are 1'IOI'IIe1'0, W, ftOf&IerO, a" n 00 lim a. ,,~-;;;- .. l)'

..- b,;'"- l)'

We prove each part separately. Recall that for S,1I ,) z, '1/ E R, ft, d(s, d(z,7I) first part, given eI> For the flnt > 0 we al80 have 1/2> ./2 > 0,80 that we ean find a positive integer N 1I such that IatJ -- a" ./2 whenever "n > N 1 can a..1I < ./2 and we can also find a positive integer N, Nt such luch that IIbb - bll.. 1' < 1/2 e/2 whenN I }, then whenever" ever n > Nle N I • If we set N == = max {Nt, (NI' N.}, whenever n > N we have

Iz -III· - 711. I.

(a + b) I(0

(a" - (a..

b,,) 1 (0 - a,,) + btl) I ... 1 1(0 On) + (b lIZ

~ la -a.I+lb - a,,1+lb ~Io

-

b,,) I btl) eI! e1 b,.1 <2"+2 <'2+'2 "'"

e.f.

This proves the tint part. The leCond limilar manner. second part, about subtraction, can be proved in 8a similar A. •a matter of fact, a few chanps M changes of lip sign in the above proof will win prove the result lor IUbtraction. subtraction. Here i. nRlt t. an alternate proof: By the third part (which sequence -1, -I, -1, is proved in the next parapaph), obeemnl obeervinl that the aequence . -1, ... (-b,,) - -b, 10 by' by the arat fil'lt pIri part .-1, •.. convergee CODver'leI to -1, we have lim (-b..)

(0,. lim (a. ...... .......

....+

b..) == = lim (a. (0,. btl) ...-.co

-=

fJ

.... + (-ba»

.... "...

(-b,.» == -lim 0,. + lim (-b ..) lim a. ( -h,,)

(-b) -

G -

. . . . ....

b.

13.

OOffYDGaN'l' • •UBNC8I CON . . . . .rr IIIIQUDC8I

49 ...

To prove the tbird ltan with the fact that converpDtlM!Cl'llDeel converpnt eeqaeDeei third part paR we Itart are bound~ get a number MER IUch that 1 '-....11< MAI and I'" 1"-' < AI M bound8!i to 1ft AI E R lOCh for all poeitive integers clOled ball elOled lit, let, inteprl ft., n. Clearly M > > O. Since a& clOl8d baU it a• clOled the precediDl &Il7 • > 0 we ... a1Io pncediDi theorem impH. impli. I-I, 101, I" 161ISM. S M. Given aD7 have ./2M > poIitive intepl' Inteler N such luch that 10 IG -- .. . .I < > 0, 10 there exiltl exietI a poeitive ./214 bit I <.I2M < ./2M whenever ft > Bence for aDY any intepr lntepr ./2M and I16b -- 6.1 > N. Hence ft > Nwe have >Nwehave

lab - a..b.1 ,..... I-IGb cab. + cab. 6.(0 1Gb .. 1tJb -- 00. ah. - ,..6.I-lo(b a.b.1 -I-(b - b.) +• b.(a - ,..), Ga) I • Slal·lb-b. +lb"I·la-a.I<M· 2M +M· 2M-" I Slol·lb-""+I"'I·lo- .. I
I '~I~I

is email clearly small if ft is larp, but we allO tunall if ft n is large, larp. The numerator i. is clear1ylllll8ll need to have b.1 in the denominator bounded away from aero, aero, or the bave the I1"'1 total fraction may not be amaIl. emall. The latter objective is euily easily accompliIhed accompliBhed by det&ill of the proof by taking" taking n 10 large that I16b - btl 6.1I < III 1/2. The formal detaila that lim 1/'" l/b. -== lib we give .. u follo..: folio... :

' ' /2.

....-

Given.• > > 0, chooee a positive intepl' Given inteler N lOch such that if ft " > > N then

IbIb!.. }. Ib - ... "-I,<min{¥, < min{~. I&!'t}. > N we have Then if "ft >

'''-'-1& 1"'1-1"

(b --"')1 "-)1

~ Ibl-'" 1&1-1& -"I - &.1 >Ibl-l&l-l&l. > Ibl-~ -~'

10 that

II?II.~ If'llt lTilt --;;;;- · .. < ,.,. (111) - ..

-....

com.,..

ThtI prov. 1/'" - l/b; ThlI proy. that lim 1/". 1/6~ To complete the proof of the ... part . . .

01 the propoeition we UI8 pan topther Jut .-It • of \lie the third pari topth.- with this Ulil ......... fol1owa: folIo...:

1) -~ Ii 1 I a. - I'~ ( ... -;;;.!!2-;;;.... Ii~l;; e

1 •• T

CI -T·

Ie II

m ......... C . . .CD ID. ImTBIC CIIa

Note the special apecial CMeI of the lut propoeition when one or the other Bequencee Gl, 01, CIt, CIt, •••• bt, .••• .. is constant: of the aequencee •• or bl , bt, ba,

-

lim (a.

".. .

+ II)b) -

- .lim a. CIa

.....

+ b,

...

a lim b., b.. , etc. etc. lim ab. ... -= G

"...

"

next very.., very eMf...wt exp..... the compatibility of CODVerpnce converpnce The Dut fIIUlt up..... of MqUeDCII MqUeDCeI 01 of real numben with order.

_ii, _it..

Propoaldoft, 1/ Gl, ai, GI, CIt, Ge, CIa, ••• CIftd c&nd ba, ha, bt, ha, are c:onHrfM' converpn' "fU"'C'I ~ Pro,.I&lon. ba, ••• Gr. fI CIftd c&nd b ,.",.aw", ,..".aiNly, -ond if CIa a. S ~ b. b., lor /01' all aU ft, n. U&eR U&er& CI :S b. fI S

0/ real number" number., toitl vitA

- . - ......-

a -- lim bIt b,. - lim CIa a. -- lim (b. (b,. For b - tJ

"... .

"...

a.). and since each b" b.. - a. a.), Ga

i8 nonneptive the theorem impli.. ia implies that the limit also is nonnegative, nonneptive. the B : z ~ 0) heiDi clo.ed. .let t (z Iz E R 01 heiDI clOIed. Dejbdtlon_ 41, CIt. CIs, Ga, •• is ia ifu:r...m, iftcrlGliftg if DttfbUtloft, A eequenae MqueDOe of real reU Dumben numbenl ai, CIa. •••• .. CIt S ~ CIa S "', ...• ~ if CIt ~ CIt ~ lie CIa ~ " ••• monoConic if it it. 11 is Cll S .. ' ,, and motIOIonic

aU. iDcreuiq inCl'8Uiq or clecnuinldecreuiq. eiU.

flrat that fll, 0.. Ge. CIa. ••• II ia • bounded increuiDI increaeiDg aequence Mquence of Suppoee ftrat Gl, CIt, reU numbers. numben. We IhaIl that the l8queoC8 sequence CODY_ converges to the limit real aball prove dlat •fI -l.u.b. Ga, ••• J. CIa for all a, - l.u.b. (til, loa, CIt, fIt, ...... ). We have •fI ~ a. n. by the definition of upper bound. For any e_ > 0 we have •fI -- e_ < 0, 80 by the definition of leut upper bound there then ia poeitive intepr N lOch that t.hat GN ON > a 0 - .. Since MIt it a• poeid". follo. . dlat Ga > a the leQuenee Mquence is ia inereuiDl, increuin&, i' it. foUowa that. if" if n > N then a. fI - _,e, 10 80 that t.hat.

CI-_
........, .... .............

EXAIIPL.. Th.lut...w, AD"y R, 101 < 1, ExAM.".. The lut reaul$ p•• Ilv. an easy proof that if CI0 E B.lol 1.

thea then

-....

Hmolim fI" -0. - O.

Pint note that 1"1-1 80 that we may IUppoee BUppoee that 0a ~ o. O. Then Firat DOte Ia-I - IafI I". 1-, 10 tllellqU8DC8G, aI,,,a,,... at, ..• ildecreuilll,ainceo· tile IeqU8DCe fl...... • •• ia decreuiDi. since a" - 8,,+1 0,,+1 =- 8"(1 a"(l - a) ~ O.

o.

I"~

II 11

The terms of the sequence nonneptive, 10 the aequence eequence is ill bounded, lequence are all nollll8Ptive, hence convergent. r= :1:. z. Then converpnt. Let lim aa a" ...

........ ..-.cD

a:J: 4Z

= art == lim (a • Cia) art) - lim a,,+l == aG lim a" 0"+1 .- . lim a" 4- .... == :1:. ~. ....

1I-.clD

.....

..-.00

.... a1oClO

Since u z, we have (a (0 - l)z - 0, 10 •Z -. -·0. a:J: -- ., O. All. GE E a,lGI a, lal > 1, then AJj a cODleQuence CODlequence of what we have Juat shown, if G I , at, ... i. the sequence 4, this fact G, GII , G ai, is unbounded. (Another proof of thia is indicated in Problem 11, Chapter II.)

I.. I'. COMPLETENESS. The definition of convergence of a sequence of pointe points in a metric apace space enables \II linut is ua to verify the convergence of a liven sequence only if the limit known. It is desirable to be able to state that a liven given sequence is convergent without actually having to find the limit. We can already do this in certain eases, that eaaeB, for the last proposition of the previous section state. b t a& bounded monotonic aeqU8llC8 always converpnt. convergent. However we sequence of real numben numbers is alwayB need a more general pneral "inner" "inner" criterion for the convergence of a sequence of pointa reuon we introduce below the concept of pointe in a metric apace. For this reason "Cauohy sequence", a sequence of pointe "Cauchy points in a metrio apace that .tiBies aatiBfies •a certain property depending only on the terms of the sequence. It will tum turn out that aU sequences and that, at least all convergent seqUeDoes sequenoea are Cauchy aequenoea for certain important metric spaces, any Cauchy sequence is convergent. ahall often be able to state that a sequence is convergent without Thus we shall having to determine the limit. We very often are not at all interested in baving computing the limit of a sequence but rather in verifying that the limit po88l!11es certain properties, and once it is known that the limit existe po88eIIe8 exista auch luch properties can often be inferred directly from the sequence. aequence.

of

D~,,'dcm. A aequence Hquence of pointe PI, Pt, Pa, Deftnlt.on. pa, ... . .. in a metrio metric apace 8pace is a Ca.y aequence aequerw if, given liven any real number number.e > 0, there is a positive integer Cauchy d(p,., p.) p..) < •e whenever ft, n, tn N such that d(p., m > N.

The number N in the definition above of course depends on on •.e. The important point is that liven CJRy te > 0 there exiBte given an" exists aotM 801M N with the desired property.

Pro""..don. A eorwer,ent ~ aequence ~ 01 oJ poi"" ProlHM'eion. pointl in G CI metric apcace apace ia it Ga CcaucI&1I Caucl" aequerw. ~.

PI, Pt,,., P then for any • > 00 th.-e For if PI,,,, Pt, ... • •• conv_ to p there II II· an &11 such that d(p, '11) p,,) < ./2 integer N 8uch e/2 whenever R n > N. Hence if R, ft, tn m>N we have d(p,., p.) d(p", p) d(p., Pa) S d(PII,

+ d(p, Pa) < i + i == e.

52 5J

111. DI. MJ:TIUC M1IITRIO SPACES IPACBII

However, not every Cauchy sequence is convergent. convellent. For example the sequence 1,~,~,~, 1,~,~, U, 31, U, ... is a Cauchy sequence in the metric space == R - (O) Dot convergent apace E = (01 (the complement of 10J (01 in R), but it is not in E. any sequence eequence of points pointe in •a metric apace space B. More generally, if we take .ny 01 the sequence which converges to a limit which is.not one of the terms of and then space, we get a Cauchy sequence tben delete the limit from the metric apace, which is not convergent. The following two easy propoaitiolll propositions give known properties of convergent sequences that generalise to Cauchy sequences. The tint first proposition is trivial 10 80 the proof is omitted. Proposition. Aft" IUbsequence 01 a CaueAy ~ i. a Cauda, aequence. ProJHJ8itlon.

A Cauda'll'equeRCe 0/ poi"'" in (J metric rptJCe i. bounded.

For if the sequence is 7"1, PI, Pa, • • • and I• il poeitive number and Ie Pl, PI, ••• i. any positive N an intepr lUob tl(p.., P.) < I• if ", m > N. theD then for any fixed '" m>N IUOh that tI(p., fa, '" the entire eequence 'P. and radius radiul seqUeDce iI Ie contained in the closed ball of center p.

max (d(P., 'PI), tI(p.. d(p., pi) PI),••••• ... , tI(p.. d(p., ,.), I). (tI(p.. N. pII), .1. Pro".,..tlon. Proposition. CM&f1ergent. ~.

CMW""""

A Cauchy Cauchyaequmce aequmce Uaat tAat Jug has CI(J ~, ~ aubHquence if it itnlf it,elf

Pl, Pt, PI, Pa, PI, •••• Let PI, •• be the Cauchy sequence, p the limit of a& convergent conveI"Ient subsequence. d(p.., p.) < ./2 if ft, '" In > N. aubaequence. For E• > 0, let N be lUeh nch that tI(p., Fix an integer m > N 80 that p. is in the convergent nbaequence 8ubsequence and 80 tI(P, p.) for" far out that tl(P, fl.) < ./2. Then for ft. > N we have e e d(p, :S d(p, <"2 tI(P, ,.) p.) S tI(P. P.) d(p., tI(p., ,.) p.) < 2' "2 2' - ..e.

•+ •

+

com,""

D.n'tlon. A metric space apace E is comp" if every Cauchy sequence D.nltion. eequence of points of E B conV8llel converges to a point of B. point.l E. proved.hortly It will be proved Ihortly that BE- Ie is complete. Other usefulexampl. UIeIulexampl. of apaceII will appear later. The lollowina followilll propolition propoeition gives giv. complete metric Ip&eel us infinitely more examples.

P.."".ttlon. ProptM'tmn.

A cloaetIlUbteC . if cloa«IlUbNC 01 CI tJ complete metric . rptJCe it II a COMplete COMplete meCrie ,""ric

8pGCe. rptJCe.

nbeet ill is of COWBe The subset coune considered to be a8 nbapace, mbspace, 80 that a8 Cauchy nbeet iI is •a fortiori one in the ori&inal sequence in the sublet oriPnal metrie apace. epaee. The result ia is an immediate consequence of the theorem of the Jut aection. reBU1t eection.

... I •.

. •

~

TMorem. R d complete.

ai, fII, CIt, .... CIa, .... • ~. be a Cauchy eequence of 01 ... Dum"'. numben. W. _ __ Let a" _ thow that tbia IUUDber. CoaIld. CoaIid the..~ leqUence converpl con'N'PI to a ... Dumber.

8 - Is ~ a. {s E R : :r: 'I S a.. for an iDfinite infinite Dumber number of 01 poIitive in...... _I.J.

fIOIIl Since the Cauchy eequence is bounded, 8 it ia nonempty and bounded from above. Therefore a - 1.u.b. exists. We prooeed proceed to prove that ........... . tu.b. 8 exiate. 01 converges to a. Given any • > 0 cbooee intepr converpe choose a poIitive in .... N such ~ Ia. _ +./2 Ia.. - ... a..11< ./2 if ", ft, m > N. Since a --I.u.b. I.u.b. 8 we have a + ./2 E 8, but ThiB means that for only aa finite Dumber in..... a - ./2 E 8. Thil number of poIitive in...... •ft is it true that a + ./2 S ~ a.., but we have a ~ .. a - ./2 S a.. for an inllnite intlnite number of •• ft. Hence we can find a apecifio apecific intepr ttl m > N IUoh aueh th&t a + ./2 > ... a.. and a - ./2 S~ ... a..,, so that 1I_a -- ... a..11~ S 1/2. ./2. Thftore Tbenlore if ,,> ft > N we have

..at

I- - .. 1-1 (a -

a..)

~I-

+ (.. - 0.)1

• • - ... 1+1 ... -a.1<2+2-"

This provea tit, ••• conv_ converpl to Go .. proves that ai, CIt, .......

For simplicity or of notation let "ft - 33;j the proof for any •ft wiD will be -.a_tially the ame. We have to show ehow that a C&uchy Pa, Pat ••• Cauchy eequence MqUfll108 PI,'" of pointa PI - (*s.,., (SI, WI, 1», IJ), PI - <-.,., -.), ,,pointe of IiE' has a limit. Let Pa (2tt, WI,.r,). ,,(Sa, positive in intepr IUCb that (:r:a, Ya, ..), ..), etc. Given • > 0 there ail. exiata a poeitive .... N lUCIa d(p., p.) < •41 if., if ft, '" m > N. Since

PIt,., ...

.. - ...,. V (s. - :r:.)' Zoo)' + (r. (,. - ,..). ,.)' + ((.. -.), - :r:.1, I,. - .. I, I~ - .. ZooI.I,. ..1.lz. ..I, 1s.-:r:..I, .. - ..1<' .,.>N. Is. - ZooI, 1,.-,.1,1 I,. - ".1, ,... I< 41 whenever ",m > N. d(,., Pa) d(p., p.)

~ Is.

'It

ThiI Thia meana that each of tbe eequencee Sa, 'II, -. 2tt, WI,,,, aad It, meaaw the sequence. -.. •••• ••• , 'I, fI, fa. ••• , .......... .., By the theoNIn tbeoNm .... theH 1IqUtIl0li .., •.. • .. it a Cauchy eequence IIqU8IlC8 in R. B7 I8qUtIIOII .... Ale converpnt, Umita by hT S",' s, ,,' feIIMlCtlveb'. NlJ*livel7. We ....... .. conVfll'lent, and we ,a.ote ,tenote their Umite prove lIIat that PI, Pa. ,.,,., PI, Pat • •• converpa point, (2:, "f, .). To do ..., .... pIOYe convera- to the poiDt , - (s, Te cia pven • > Q 0 cboole intotpr" __I, choose N such auch that for any intcw .. > N we .... haft Is - Sel, I, - ,.1, I. - .., e/...;T. Then for" I. .. I < ./'\IT. for .. > N we have bave

we have

d(p, 51 + G(J d(P, ,.) p.) - v'(s - .. s.)' ~ ~

,.5' .,. ..5' + (. - ..,. -,# .. , -,..
-

'11lia proves that lim ,. Thia provee p. - p.

16 "

III. METRIC 8PACU BnCEI!

,5. t 5. COMPACTNESS.

DeJinition. apace E is compact if, whenever 8 is Definition. A subset 8 of a metric space contained in the union of a collection of open 8ubsets subsets of E, then 8 is contained in the union of a finite number of theee these open 8ubsets. subsets.

. Clearly any finite subeet IUb8et of a metric apace is compact. Unfortunately give a 1_ I. . trivial ex&lnple example of a compact we have to wait before we can live IUbaet of a metric apace. space. An example of a noncompact 8ubset subset of a metric lpace ii, however, easy to live: give: the open interval (0, (0,1) apace 1) is not a compact apace ft, is contained in the union of all IUbeet of the metric space R, aince lince (0, 1) iI ,.. is a positive integer, but iI is open lubseta lubsete of R of the form (l/ft, (1/", 1), I), where where" IUbeete. It not contained in the union of any finite number of these open IUbaeta. will be seen 88 go along compactn. . is a property related to comwiD as we 10 aloog that compactoesa . ., but much atrolller. stronger. pleten pleteness, The definition just given of aa compact subset of aa metric space can be applied to E B itself: the metric ·apace apace E ia is called compact if it is a compact 8ubset lubeet of itle1f. itself. This means that whenever E is the union of a collection of lOme finite 8ubcollection subcollection of these open open subsets, it is the union of some aubeeta. subeete. 8 be an arbitrary 8ubset subset of a metric space E. When we consider 8 Let S 88 a lubspace IUbspace of H, E, an open ball in 8 is simply the set of points pointe in 8 of an as open ball in E whOle whose center is in 8, S, that is the intersection with 8S of an open ball in E whOle whose center is in 8. S. ThUl Thus the open subsets of the metric space 8 S are precisely preciaely the intersections with 8 of the open 8ubsetl subseta of E. apace Hence S 8 iI is •a compact subset of E if and only if the metric space 8 i. is compact.

ODe One example of •a compact metric apace space will give us many more, by me&nl of the fonowinl following result. reault. me&DI ProJHMition.

AftJl . . . 0/ G A"y cloMl cloted ...w.t CI compact metric .pace it compact.

For let the closed c10eed lubset 8S of the compact metric space E B be contained say 8 C U U" where in the union of a collection of open lubseta of E, I&y

. .

,., iEI

subaet of E, i,ranging indexing family 1. Then each U, is an open IUbeet ranging over an indexinl E C (V (U U,) U U e8. Since 8S ia is cloeed, closed, e8 is opent open, 80 by the compactn. compactn. .

,

of E B we can find •a finite 8ubset lauch subset J C 1 luch that E C (U U,) V U es. e8. Hence lEI 'EJ

8 C U U,. Thillhcnn This Ihowa that 8 is CDmpact• compact.

"

• 1

Here an lOme of the bulc hulc properti_ of compact seta. Bere properti. 01

p........,..,..

. . . 0/ G apace it i8 bounded. 1,., ~. A emnpad compact . ..w.t CI mItric metric .,ace In particular, •• . NCrio " . . it . ...,,_ . , ...".. IpGQJ ia bounded.

I"

OOIIPAOl'NUI

IS

tpaCe is the UDion union of itl ita open balla, and the union of For any metric ip8C8 any finite number of balls bal.. is a bounded set. Bet.

Propoa'tfo" (N ..,., ••te property). 1M 8., 8., ••. . •. be Prop08'do" (N..,., LM 81, Sa,

CI aequenu ~

oj

ttememplJ ftOMmptr cloIetI aubaett oj CI G compact metric meCric 'qoee, 'lPG", UIitA tIIitA 1M tAt propt;rIfJ proptJrly tMt that 81 :::> Sa 8.:::> •..•• TAfn tMrt 81 8. :::> 8.:;) tAm .. ia ,., at ,." one OM point IAae that bdtmg_ fIeltmtI. to fl/IClA eGOI& oj 0/

T_

tAt . . 8" 8l, Sa, Sa, ••••. 1M .". mUlt hAve If not, we mUit bAve

f"\ 8. 8.... thAt f'\ - {lJ, r;J, implying tbAt a-l........ .-1,1,1....

V U

...,1 ...... . - l........

es. is e8.

the entire metric apace B. Since B is compact it is the union of a finite number of the open aubeetl IUbeeti e8l, e8a, •••. Since C e8a C e81, e8., ••.. Bince e8. e81 C ..• , we mUit mUlt hAve e8. for lOme n, whioh producel produo. the contradiction contradiotion ••• bAve B - &S.

es.,

es. es.

{lJ. S. - r;J.

propotition doeI DOt not hold if the word "compact" is ia replaced The above pIOpOIltion by "complete": for example let B - a, III e II ~ n I, R, taJdlll takilll S. - 1* E a R :: * n -1,2,3, ....

~ '--

FJQVJUI 1" J'JG1JUI1..

N.ted '" . . JII'OP'R7'. propeR1. TbIN TheN II a JICIia' point 00IIIID0Il the ICllIaI'8I, aquaree, each of 01 N.t.ed 00DlIDGIl W too all tile wWeb wbieb)au .... W.tIIe ball ,the dJIDIDIIoM cD........ 011... 01 itAI pncllaullIr. pndeoeMor. (A oIoIed oIOIId IqUlft Iquare" II oomc0mpact, by b7 •a t.beonm w be proved 1hort1y.) Ihortly.) tbeorem to

pac"

For a& better inaipt into the meanilll meanina of compaotn8lll, compactnelll, we introduce another definition. ~"'do".

If E H is a metric space, 8 a subset of H, H, E, and p a point of E, then p is a cluatfr cl,..,. point 0/8 if any open ball with center p contains an.infinite number of points pointl of 8.

16

m.

MJITIUC MB'I'IUO IIPACB8 IIPA0B8

An infinm tuba. tub,et oj Theorem. A n infinite clu,ter point. cluater

II (I

compllC' metric compoet

epGU IptXe

1Iaa klllt one 1&aa lit at leoa'

If this were false, then for any given point of a certain compact metric baving the given point as center and space E we could find an open ball having containing only a finite number of points of the infinite subset 8S of E. E is the union of all such open bans. balls. Since E is compact, it is the union of a 8 is finite, a contrafinite number of such open balls. This implies that S diction. ,equence oj points in a compact metric IptXe apace hOI hili Corollary 1. Any sequence ntbsequenu. vergent aub,equence.

II (J c0n-

Let PI, PI, pa, .•. th~ sequence, E the metric space. We must Pt, 1'" ••• be the as the set 11'1, Pt, Pa, pa, ••• .•• J\ ill is infinite or not. separate cases, CaseI, according 88 (PI, PI, CGle tPI, Pt, Pa, CII8e 1. The set 1 pa, ••• .•• J\ is infinite. In this cue case the Bet set 8S - 1 PI, 1'1, Pt, Pa, pa, .•.• bas at least one cluster point, IBY say l' E E E. Pick a posiIPt, • • }) has tive integer ftl p and radius 1, nl such that 1'''1 1'", it is in the open ball of center l' nl such that 1'.. 1'.. iI is in the open ball of center l' then pick an integer tit tat > ftl p and radius 1/2, then pick an integer '" tit > tit .. is in the open ball of radiUl At euah lUoh that 1' PAl II In l' and radius 1/3, etc.; etc. j this procell process can be cOntinued indefinitely center p l' contains an infinite number of points of 8. since each open ball of center 'P pa, ... whose n n'"'A term has disWe end up with a subsequence of Pt, Pt, "3, less than l/n from 1', n -- 1, 1,2,3, subsequence tance lea 'P, for all ft 2, 3, ..... . •. This subeequence converges to p. Cllle Pt, PI, Pt, 'Pa, Pa, .•. Cale I. The set 1 (PI, ••• J, is finite. In this case at least one E {Pt, 1pa, PI, Pt, 'Pa, pa, ••. point l' E ••• }\ occurs an infinite number of times in the sequence PI, Pt, PI, p, 1', p, 1', p, ..... pa, .... •••• Thus the convergent sequence 1', ••• is a subsequence o( the given sequence. of

Corollary J.

A compact compllC' metric IptXe 8fXJC6 it ia complete.

For any Cauchy sequence has baa a convergent subsequence and therefore is itself convergent.

't

Corollary!. Corollary

compad tub.. tubeet oj II epGU it ia do_. cloHd. A compod (J metric IptXe

convergent sequence sequenae of pointe points in the compact sublet eubset muat must For any coDverpnt its limit in the compact sublet, subset, by Corollary 2. Thus the theorem of have ita , 3' impliee §3 impliEll that the IUblet subset i. is cloeed. clOl8d. We now know that any compact subset of a metrio space is both closed and bounded. We proceed to prove the fundamental (act fact that any closed bounded subset of 1P E- is compact.

£em..... oJ B-. B". TAM Jor . Lemma. La Ld S S be k 0 botmcW bounded eu6M aubM 0/ TAM/or ., . •f > 0, 8 U " ... ..... lea. . ia CAe wiota ""iota 0/0 ftail,e "umber aUMbtr oJ'" 0/ eIoNtl NUt oJ,..,., 0/ ...... Ioifted ita 1M oJ (I ft."ite •. We belin the proof by shoWinl aboWinc that tba& BB" itlell of .... itaell it the union 01 a '" 01 of c10aed balla illustrated in Fipre Fipn 16. 15. evenly spaced closed balta of radius "I, •. . iDuatrated

",-..." V /'" I?"'..

I/

I/

I........, /i

k(

. A/

....... N

.7"""'.. V""'-..

....... ""[\ 1\ l) r\

r--.... :""'00.

--..... ~) ~) ~ "-...... ....

I,(.

~

t......'\.1/

( 1I/~/1:' -........ J K ......... [I r-.... ........ ~

K

~ ~I....o' r..,...o'

i'o.. ,

.-/ ~ ,r ~

[) J

J II

\ ,

Ij

1'..~ 1-......... .--/

To be conerete, concrete, eolllider coDllider the pointe in 11" E" of the form (ot/-, (0./"', Gt/., tItt/-, .•••, ~ ., 0./".), /It, ••• , a.. CJ,. are intepn a./"')' where 01, Oa, lit, integen and ". '" it is •a fixed poeiti. poeitive. in...., to Ihortly. For any (z., (Za, s" St, ..•• •.• , ~) s.) EE 11" E" &here there are in.... in.... fA) be determined shortly. CIt, fit, a.. IUch luch that ... Ot, ••• , CJ,. 01+1. S~ ~ <, 1 - 1, J, ••• , a ;-01 ~ --.--' i-I, 2, ..•,,,

..

..+1

'"

'"

In and we til-. theIl have baft (by LUB. of Chapter II)

--------------------a«SI, +... ";.)'1 a(SI' ... ,z.),(::, ,s.),(::, ..., ... , :)) :: )) -- ~(Sl--;-)' ~(Sl-';-)' + ... +(z. +(s. -- :Y _I..!.. + ... +.Jr+..!.. _ "'" < _1..L+ v'ft.. "I ".' 'I",.

"...". m' '"

II •

m ....... c ... ..ACa 10. . . . . . .o C.

Now IUppoee that the fixed intepr '" tn bad been choeen greater than Vfi VftI t,f, that Vft is contained eontained in the cloeed closed ball of Vfi1m I"" < f.t. Then (ZI, (Zl, •••• , %a) Zoo) iI radiUl (tJalm, ••• , ..1m). B- iI is the union of .u all ndiua et and center (a,J"'" a./",,). In particular g. theM balla, for varyilll intep'&l ClI, CIt, ••• Ga. It remaina to ahow that if varyiDl intepoal o. 0,, ca.. 8 C Efinite number Dumber of valu. of Gl, (II, CIt, Ga, •• •••" , ca. CIa IUfBoe B" i. ia bounded then a &nite to live whOle union oontaina oontaiDi 8. B. To do thia tbil note Dote that pve UI ua balll balla of radiUl ndiua •t w'" IiDce oriPn. Iince 8 ia bounded it II contained in lOme ball with center at the oriJin. AI, then for each point (Zl, (Sa, *S, St, •• •••0,,~) S we have bave If thiI ball hal baa radiUl ndiua JI, z.) E 8 1 1 I v' Z1 Zool S JI. Since S v'Zi z.I z.1 " + ~ + ... + Z-I M. '~I "ZII + Zli + ... + Z-I for i-l,2, ... (ZI,~, ••• , Zoo) Za) E 8. If, H, .. .. 0,, ft, n, we have I~I S M JI whenever (Zl, *s, 00', above, til, luch that oJ"" oJ- S Z, Z4 < ('" (iii + 1)/m, Gl, GI, CIt, ••. , a. ca. are intepn IUCb 1)/"", we have

10

0

0

+., + ... +

0

.,

"

-it

,S.,

1z,1

+ + ... +

o •• ,

I: I-Iz,+(: I-I~+(: -z,)ISIz,I+I: -~)I~I~I+I: -z,ISM+!. -~I~M+~. HlIlce _, ... ••• , z.) E 8 we have 1 I",I ~ I S mAl mM + 1, awiDl ahowilll Hence whenever (~I, (Zl, *t, that there will be only a &nite finite number of poIIibilitiee po88ibiliti. for the intepn intelen GI, fII, •••0,, fIa. is completely covered by a &nite finite number of our clOled cloaed ca.. Thus Thua 8 ia .... ofndiua baUa of radiua ..

-it'" ..

na.or.m. A"" An, cloaetl clotetI bouftded bouftcIetI aubM g. if T"-rem. __ 01 .. i, compact eotnpad.

,aile

8uppoee the Uleorem cloaed bounded IUbeet nbeet ~rem and that we have a clOled ia Dot not. COIDput.. TbeD t.bere 8 of IIB- that iI compact. TheIl there ia a collection of open IUbeete aubeetl lUI - I Ud., eontaiDa 8, but IUch meh that the union of U,J., of .II-. whOle union oonWM DO UJ contaiDa is bounded, the lemma no finite &nite IUbcoDection IUbcoUection of IlUI oonWM 8. Since 8 ia ..... ua that that. 8 ia contained in the union of .. te1II UI 01 a &nite finite number Dumber of cloeed dOled bal balla ndiua 1/2, .y say Bit B., .•• ••• ,I &. of radiua B a, Sa, Bpe Thua ThUi

w'"

, f

8-

.. · U V u···

(8(,,\B~V(8(,,\BIJV (B"B~U (Bf'\Bd

(8("\ Br). (8f\B r ).

leut one of the I8tI 8 ("\ Bit ("\ B., At l. .t ODe leta 8" Bl, 8 f\ BI , •••• , 8 ("\ f\ Br, .y 8 ("\ f\ BI, BI , is it not. not r , .Y any &nite contained in the union of ailJ finite IUbcoUection of lUI. I U I. 8 ("\ f\ BI iI is clOled cloaed t.he interlectioD intellection of cloeed clOIed 18tI) leta) and if 71, (heiDI the p, ,q E 8 ("\ f\ BI BJ then d(p. d(P, ,) q) S S1 (linea each 01 of the pointl pointe 71, baa diatance moet 1/2 from the center of (linee p, , hal distance at mOlt t.he ball B B/). 811 -- 8 ("\ B,. Then 8 11 ia clOled the S" B~ Then cloaed and bounded, not. not contained 1). Set 8 8nite IUhcoUection lUboolIection of t U I, and if 71,' p" E 81 then in the union of any finite d(p, ,) S 1. Now apply the the1emma tl(P,9) 'lemma to 8 1a and t - 1/4 to pt 8 1 contained a finite &nite number of closed clOIed baUa balls of ndiua radius 1/4 and repeat the in the union of & aJ'IUDl8I1t to get pt •a cloeed sublet IUbeet 8i of 8 1,1, not contained in the union above argument &nite aubcoUeetion IUbcoUection of I(UI, U J, lueh lOch that if 71, p, ,q E E 88.1 then d(p, ,) SS 1/2. of any finite RepeatiDl the &rIUment arpment with IIe -- 1/6, 1/8, 1/10, •.. we obtain a aequeoce Repeating sequence clOlled leta eetI 8::> 8 :::> 8 81 :::> 8.1 :::> 8.::;) 8.:::> •• · ., none of which is contained in ·the the of closed ::;) 8 union of any finite &nite lubooDectiOD IUboollection of UDion 01 IlUI, UI, and .uch such that if 71, p, ,q E 8" S. then d(p, ,) :S 8 N ia il empty, 10 80 let 71" PN E E 8", 8N, S lIN, foreachN for each N .. - 1,2,3, .... No 8" 0

0,

0

0

0,

0

•

N - 1,2,3, JIa, PI,,., .•• a CauchJ Cauch7 1IqUIDCe, leqU8Dee, IiDce IIince 1,2, a, •... .... The aequence eequence ".,,.,,., ••• iI ia. aDd ., II, fJ _ > N the til. ,., ~ p, E BII, 8., 10 that if a, fJ. N are poeitive PQIIitive intepra aad "(p.,~) E- ia is complete, leqUeooe "., JJ&,,., COD"(P., piJ ~ lIN. Since 8io.ce B" oomplete, the IIqUIIIOe PI. Pa, ••• eo. yers. Pa, PIt PI, ", Pt. ••• ia is in the cloIecl cloeec:l YeI'I- to • point PI of E-. B". Since each point polnt "., .t Po E 8. Hence ia in ODe ODe of 01 the ... IetI of the ooUection collection . 8, we have aIIo PI Bence PI II (UI Po E U. tU). Since U. Ieia open there IeII aD an open ball of eenter center PI Po ( UI,, .Y lAY "E u. E (U). and lOme radius U.. Pick 'lOme N 80 radlua • > 0 that is ia entirely contained in U .. PioklOme larKe ,/2. Then for aDy any point I' PE 8N we larp that lIN < ./2 and tlc,., tI(po, PH) < ./2. E 811 have • 1 • • d
CONNEcrEDNESS. 16. CONNECnDNESS.

The intuitive-idea conneotedn. ia ia·aimple enouah. It ia is W'fIt.n.ted ill'fltrated intuitive idea of oonneotecme. Iimple 1IIOuP. in Fipre Fiaure 18. 16. But of coune we need •a precise definition.

Flau.. 18. 11. A, S, B, C, A V FloUD U C, B V U C, A V UBV U C are eoDIIeOtAld. eooaected. AVBiaDOt. AUBiaDOt,.

~don. A metric 8pace apace B ia conRBClBtl ~ if the onlysubeeta DeJinldon. E is only subsets of B E which clOlled are E B and fd. B of a metric apace is •a are both open and cloeec:l ~. A subset 8 conMClMl subset aubaet if the subspace 8B is connected. CimRBClBtl

60

m.

IOTBIC BPACBS IOITIIIO BP.lOU

Thus baa a• aubeet subset A,. A ,. S, B, IlJ ThUll a metric apace E that i. is not connected hal that is both open and closed. H, IlJ fIJ is both open and clo.ed. closed. c1oeed. Then a1Io aIIO eA SA ,. S, Setting B - eA, we have E =-.A Bare diljoint nonempiy nonempty -.A V U B, where A and B are disjoint open aubsetB bewrittenE A U B, aubaete of E. ConverselyifametrioapaceEcan Conversely if a metric spaceE can be written E -.. AU where .A IUbIeta of E, H, then E is not coneonA and B are disjoint nonempty open aubaete nected: for A ,. flJ ~ E (ain!ltl (ain~e A and B are disjoint and B ,. 1lJ) fd) IlJ (given), A ,. and A is both open (given) and c10eed closed (aince (since ite ita complement B is open). a,",eER, If tI, II, c E R, we.y we .y that ce ia it ....... ""'-" a 0 GrId" aM " if either a 0 < c <" < b or

b cJ) 8 - (8r\ (~E R : sz < e) (~E el) expresees nonempty open aubaete. subleta. expl'ell88ll 8 88 the union of two disjoint DOnempty It will that any c10eed closed interval in R i. is connected. In wi1I be shown .hown shortly .hortly tbat the next chapter it will wiU be eeen aeen how to deduce from this fact that certain sp&ceI, for example balla other metric &paCes, balls in ga, E-, or the aubaete sublets A, B, C a of Figure 16, are connected. The followinl is one of the principalal'lUmente principal arguments one UII8I us. to prove theBe and other connectedn_ connectedneBI reaulte results for aubaete sublets of ga. Ea.

Pro,...dora. 1At Lt1C 18,)., (8.1., ". a collectiofI collet:Iion 0/ ProJHMidon. be CI 01 conneIlCetI comwdetl ,.",.,. aub.ca 0/ 01 aCI tneCric fMtrie .,.. B. BuPJlt* ~ IMre lAIN . . " it E 1 . - IIuJt fur eaeA i E 1 "., apatA E. IUCA t1uJI, lor U&e .", . . 8. GrId aM &, . late nonempe,..,.,--. TAM U V 8. &. . Ga noMmP'1/ iftterlfJdion. TAm 8, ia it ~ cormerJed.

.,

., fEI

f., .

8uppoee IUbeet. A Suppoee that 8 - U 8, 8.. ie is the union of two disjoint open aubeetl

mUllt show .how that either A or B is empty. For any i E and B. We must E 1, B. - (A" (A r'! 8,) V U (B" (B r'! 8,) B.) exp . . . . the connected I8t B. 88 the union of 8, exPreIIeI conneeted let 81 .. disjoint open aubHte, 8. and B r'! jUllt the IIetI two diljoint IUbletl, 10 that A r'! ("'\ 8, ("'\ 8, are Just seta IlJ in some lOme order. Without Ie. 88Iume 8 i , J2J 1088 of pnerality generality we may . .ume tbat that r'! 8_ B.. - 8•• B... Then A :J :> B., 8 .., 10 that for each i E 1 we have .A A r'! An () 8. 8, ,. ,.. 1lJ. Ill. ThUll A r\ r'! 8. 8, - 8, 8. and B r'!8. r'! 8, - SlJ Thus () 8, - SlJ. flJ. Since Sinee B B" J2J for all i E 1 we have B .. == SlJ, /lJ, proving the proposition.

8.,

R, or any tiny open or cloHd Theorem. H, cloNtl interval in R, ia if etmnet:Ced. cormerJed. IOmewhat more leneralltatament 88 easy A somewhat general statement is just .. euy to prove, namely aublet 8 of R which contaiDI contains aU that any subeet all pointe between any two of 01 ita points is connected. To prove this, point. tbiI, auppoee IUppoIe that that, lOch lUob alUblet alUhlet B 8 is om not conCOD-

nected, 10 that we may write 8 - AU A U B, where when A . . B an __ nected,1O Md IN cIIIJoIat dIIjoIat .... opeillUbeeti 01 8. Chooee ChooM a_ e aacl ......, empty openeubeete of 8. E A, be I> E B and ........ -u we 1Ila7. 1Da7, b. By ueumption (_, A" B" b). that a_ < 1>. (a, 1»b) C 8. Set A& A. - A f'\ (., (a. b). &). B. - B f'\ I., (a, 1». A., B B.1 are disjoint open au"" IUbeete of (_. bJ, a_ EA., EA •••&E Bit Then A•• (a, 1», B.. and (., I., .1 &) B•. FlOm From theM facta we derive. A. U B•. theee facte derive a contndiction, eontndiction, _1aUCnrI. u fo1IowI. 8IDee 8iDee B. ia an open aubeet IUbeet of (_, b), ite ita complement A& .... . . . 01 (G. (a, 6], A• iI • . a .cloeecl ...... Ia, .1. &). clOI8d aubeet 01 hence •a cloeed of R. Since A& A. Ie ia aIIO aIeo """8"\P*)' DODeDlP'1 ... aDd bouacIed bouDded 110m fIOID nault of 12 tel.. tella us u. that ..............t~ cia .. above, the lut result \ba~ theN theN.a c ill A A.. Bu~ Iince eince A. is of (., (a, .1 h) Since 6 bE B B.1 we must have c < 6. b. But Ie an . " . 01 it must contain the interlection la, 6) IOIDe OJMD optD ball in R • 01 . . . intenlection of [., b) with lOme c. mu.t contain pointe point. peat. peater than c. Tbilil c, hence it must Tbi8 • 00Idndictl0D., contradidioD., ... aDd thie proves the theorem. Ulia pIOV.

.a

10"01_.

PROBLEMS

1. Verily the.follcnriq .... metric epaoee: epuII: Verify that theloUOwiq (.) Dum~ with (a) all II-tupl. n-tuplee 01 real num~

d«Zh •.. , z.), <-a, II«ZI, .". z..}. (,s,

?:t

..., r.» -- ~ Iz. I~ -.,.1 - llel "., rJ)

(b) aD bounded infinite eequenoee s - (Sa, ......... ) of ....... wltb iDbite lIqUtIDOIIa 01 . . . . . . of 01 .. a, db

d(s,,) II(a,r) - I.u.b. I.u.b.'t ISa -

,,1,1%1-,,1,1.. .. I.IZI- ..I,I.. --,,1, ..I, ... 1

(e) (B. X 1ft, Bs, 11). (Bb (Bs, cit) .... me&ric II alWD (c) (BI X cI), where (B cia), (1ft, 4) ... metric ..,..,. ... I" .. pYa I , cit), d«Sa,~, <-a,"') - DIU (cia(.., ,a), 4<.. II«Ift.~, (,s.r.» max ,cIt(Ift,N,clhs.r.lJ.

,.,1.

bJ' b7

2. Show that (R X R, cI) • is •a metria epace, when where R. II) metric .....

II«z. r), V), (s', (z', r'» d«z, r'» -

I,I+Ir'I+ls-s'l .... Irl + 1..1+ la - rl 'ih 'ih'" r {{ Ir-,'I Iv-v'l Ha-r. 11$-".

..... ....

IDUltrate by diaaralll8 diapama in the plane 1fI If' what the opeD balII 01 tIdI lUuatrate baUa of thia ......... ........

3. What .... the open and cloIed in the metric 01 . . . . . (t), l? oJoeed baDa baIJa iD metria .... of (4), It If Show that two balla balJa 01 differeDt dilereat ceoWra ceaWn . aDd . ... . .. BIlow . ndii may be ....... What Wha&. open leta in this metric apace? openI8te

PYfID by (Sa, I(Sa. z.) Sa > %II ZlJ .. II a,.. 4. Show that the IIlba IUbeet of 01 If' BI liven ~e E 81 BI : .. opeL 5. Prove that afty any bounded IUbeet of .,. ~ bounclecI open opeD IIlben 01 ... • ill the UDioD UDioa of 01 cIitjoiat cIiriIaIat ......... vale.

E.:.. . -

G. Show that the IIlbiet aubeet of 1fI I, .. > 01 II ..... .... 8. If' liven PYfID by (Sa,'" '(Sa, 1ft) e 81 : .... - 1, details of the proof of 7. Give the detaila 01 the ... Jut pnpoIltiaa propoeitioD 0111 01 II ............. f ...... from fl'Olll below. 8. PIoYe ............. Prove that H the pointe 01. ofa _ ClOIlvera-l ...... 01 paildlia poiJda Ia . a . . . ....... . .... reordend, reardend. tbIo tIIIn the .......... ..., ...._ -ftIIIII ... _ ~ tit to .... tile _ ....

61

m C118 W .........o ........O ••• . ..t.OIIa

l...im,. -, lim,. -, .....

9. Prove that lim 'P. - 'P in a given metric apace space if and only if the aequeac:e sequence Pl, p, ·•.. · · is convergent. Pt, "p, Pt, PI, 'P, " Pa, PI,',

-

10. Prove that if lim p. =- p in a liven set of pointe points given metric space then the Bet (p, ,., •• I is closed. I", "., PI, Pa, PI, •••• c1oeed.

Bhow that if ., Ge, ••• is a 11. Show CIt, ., CIt, Ge•••• a aequence sequence of real numbers that converges to ., 0, tbeD thea

12. Prove that the eequence aequence

%a, •••• Z., %I, z.. Zt, ••

SI,

.. 1 and of real numbers given by zSl...

z....1 Zo.+. -- s. z. + ~ for each" :. - 1,2,3, ... is unbounded. s. z. 13. 18. CoDlider CoDIider the eequence aequence of real numbers

1

1

1

--I' 1 , .... i'j' --1' 1'···· 2+2+- 1+:I2 2+! 2+ 1

2

Show tbat ita limit by fint first IIhowiq ahoq that tII&t thia this llqUenC8 lICpIeDce is conYerpD" converpDt and find ite the two 18queDell 01 alterDate terma are monotonic and findinc finding their limite. lIqueoeea at 14. monotonic IUbeequence. aubeequence. (Hine: (Hint: Tbia This is 1'- Prove that. tUt UlJ &D7 IIquence IIqUeDCI8 ill in R baa a monotoDic elIiIta a 1Ut.equence .., if there aiata aubeequeo.ee with no least leut term, hence we may BUppoee &11M each eICb IUbeequenee IIlbeequeDce has a leut term.) (Note that this tIIa\ thi8 result and the theorem bouDded monOtOnic leQuences sequences gives another proof laD OIl OD the CODverpnee converpDC8 of boUDded tII&t • it II complete.) comp......) dIM

11.1MB B is called an ir&lerior 11. 1M 8 be •a 1Ut.t subeet of the metric Ip&Ce B. A point ,'P E 8 interior ,... of 8B if there is an opeD baD in B ,.., open ball E of 01 center eentM ,'P which is contained in B. 8. Prove that the set - ' of interior points pointe of 8 is an open BUbeet subset of B E (called the iMriIr 018) of 8) that contains w.ior contaiDa all other open subeete 8Ubeeta of B E that are contained in 8. 18. Let 8 be a 1Ubeet. sublet. of the metric epaoe B. Define the cloauu cloture of', of 8, denoted;9, denoted!, to be the iDtenect.ion intenection of all ebed IUbeete of B e10eed subeets E that. that contain ,. S. Show that (a) 'B::> J::> 8, and 8B is Ia ebed oaly if J cloeed if and only B-- 8 J iI is the limit. of eequeoees (b) 3 eet' of all aIllimiu eequencea of pointe points of ,8 that. that converp converge in B J if and oraly if any ball in E of center ,p contains (0) a point ,p E B is in 3 aDd only conwDI pointe of 8. if , ill not an interior point. point of e8 " which ie is true if and only if, 11). (cl. Prob. 16).

(ar.

BUbeet of the metric apace B. 17. 1M' 1M S be a subeet E. The bouradcarr bountlaru of ,8 is defined to be "iI (d. (of. Prob. 18). Show that J niB (a) B II is the disjoint union of the interior of "S. the interior of e8, es, and the bouDdary of of ,8 bouadary Ia cloeed c&o.cl if and only if ,8 contains ita boundary (b) B ia opeD if and only if ,8 and its boundary are disjoint.. (0) B is OpeD disjoint.

18. If

G•• G"

Ge.... 01 real DUIIlhIra, GI, Ge, ... ••• ill a bouaded bouDded ........ of awnben, cIefiDe cIefiae Iim lim .. ,.-IllPP"a..IIllqIleDOe

--

(also a.) to be (alao denoted IIiii Giii a-)

I.u.b. Is a- > s iDfiDite number of intepn "I at l.u.b. (~ E e •• : .. ~ for an iDfiDitAlllUlDber defioe lim inf .. (also denoted Jim a.) and define a- (alao Ga) fA) to be

--Ge.... ...

I~ e. <:r;z for an infiDite DUmber .• g.l.b. Is E. : .. a- < number 01 of intepn integers "I. at. &Up ... Prove that lim inf .. Ga S ~ lim sup Ga, with the equality holding if and ooIy ooly if the aeqUeoC8 CODVerpa. lleQueace converpL

AI. CIt, Ge, • •• and 6.,"', 6..... ba, ba. ••• be bounded aequeocea 19. Let G" lleQUeDces of real numbers. Show that lim sup ~ lim IUP" IUP Ga + lim 8m IUp IUP ba ba,• &Up ((Ga .. + ba) S

.

........

~ .~

with the equality holdiOC holding if one of the oricinal aeqU8DC88 CODVerae8 originallleQUeDce8 convergea (el. (cf. Prob. 18). oompIu DQlDben 10. The complex numbera C - • X. (el. (of. Prob. 8, Chap. a.ap. II) are the UIIIlerlyins underlyinc 01 the metrio . . of metrie ....... .... P. The IIMItric metric in .. P therefore ~ore makee C C iteelf • metrie ..... ..... If If •_ e the 116IoI. ",.,.." raI. 01 by I-I, I_I. isill defined metric e c, the of "_, cIeaoted denoted by defined by by d(_. 0). Show that 1_1- "(_,0). (a) Is I~ + irl 'wi - ~ I_I agrees with the prefl+iii if :r;, z, rWe. E R (and therefore 1'1 vioully de6ned de6aed 1' I_I1 if, if _ E e .) vioully R) + ..1SSI-.I+I .. I for all ", (b) I" I., +.., + 1..1 '" .. e EC (c) IUtI1111' 1..1for all IIla,... I.<l - 1'.1' .. e E c. C. 21. Show that the proposition page 48 remaiDa true if the word "real" is ill repropoaition of pap remains Vue placed by "complex" "complex" (of. Prob. 20). 22. A ftCIf'IIIecI NCIor ."... qcace is a vector apace V over •• ., fA)pther together with a real-valued ftCIrInICI ...... veofA)r epaoe cal1ed the IICIf'If& norm and indicated by UH, Uu at AllY any function on V, called AH. the value of 01 • H element s~ E e V bein& beinc indicated iDdicated UzU, 8:r;H. having the following propertiea: (i) HeA fCit all s~ e Usl ~ 0 fClt EV (ii) AsO - 0 if and only ('ai) n~D ooIy if s ~- 0 (ill) Rc:zH -lei' - lei' Os. z eE V and AllY any e e. E. (iii) Res. R~ for any e (iiii) lis rH S~ lsi fta:n + OrO for any s, r Ee V." V.II~ + W. ~. W DOl'IDed vector vecfA)r epaoe Show that •a DOrmed apace V beoomeII a metric apaoe IIp8G8 if for AllY any s, rl. ReoaIl . of of real ~. rev We V we take "(s, d(~. r) W) - Os De - ,I. Beua1l that the . Bet 01 all .tuplee n-tuplea 01 numbers is a vector apace acaIar multiplication numbera epace .- over • if .ddition addition and ecaIar multipHcation are de6aed by defined (~., .... eJ +
x.

".1

-

nrll

n,n

"w

64

III. METRIC MJ!lTIUC SPACES SPACBS

normet.l rep1acinl normetI IJeCtor tJeCfDr lpOCe, tpaa, got by altering the above definition by twice replacing the symbol R by C.) 23. Prove that if V is a normed (ef. Prob. 22) and 0., 01, lit, fit, CIt, Oa, •••• nonned vector space (el. •• and It, la, bt, 6"bt, ..• .•• are convergent sequences l8qUences of elements of V with limits 0 and "b respectively, napective1y, then lim (a.. b..) - a + b 6 and lim (a. (0. - ".) b,,) - 0tJ - b, 6, (Iloo b.) ,,-. and if furthermore i8 a ..equencc converging to c, furt.hermore r1, rio Ct, el, CI, el, ••• is I'ICquenee of real numbers convergiq then lim c.a. c"a.. .. - ('a. ro .

..... "...

+

....

°+

.

~.

24. Bhow subset. Show that a complete subspace of a metric space is a closed subeet. 25. Write down in all detail the proof that E· is complete. (H,"': (Hint: A convenient •.. , %.(").) notation is p. - (:rl(·), (s,(·', ... s.(·').} 26. Find all cluster points of the subset of R given by

{~ + ~ : fl, "', m ...= positive integers}?:I. ?:T. Let 8 be 8a nonempty subset of R that is bounded from above but has no great-

l.u.b. 8 is a cluster point of 8. est element. Prove that tu.b.

28. Prove that a subset contains all its subeet of a metric space is closed if and only if it contalna pointe. cluster points. Let 8 be a sub8et subset of a metric space B E and let" Bhow that p is a cluster 29. ~t let l' E E. B. Show point of 8 if and only if p is the limit of a Cauchy sequence of points in

8" elpl. 8"elpl. 30. Give an example of each of the following: subeet of R with no (a> an infinite aubeet DO cluster point (b) a complete metric space that is bounded but not compact (c) a metric space none of whose closed balls is complete. 31. Let 0, 0,"b E R, 00 < b. The following outlines a proof that (a, b) is compact. Rewrite thi" thill proof, filling in all details: Let I U,I,el U,lle, be a collection of open subl!ets of R whose union contains [a, subsets (0, b). Let S - 1% E (a, (0, b) : sx > 0a and [a, s) is contained in t·be u.nlon of a finite number of the sets U,l,el (0, xl t.he u.nion 8ets I(U~ JiE' I. J. Then U, for some ii E 1. Since U, l.u.b. S E U. I.u.b. U i is open we must have l.u.b. tu.b. 8 -=- b E 8. 32. Show that the union of a finite number of compact subsets of a metric space is compact.

I U,I,el U.) ,e I a collection of open subsets 8Ubsets of E B 33. Let E be a compact metric space, 1 whose union is E. Show that there exists exiata a real number ft > 0 such that any closed ball in E of radius. is entirely contained in at least one set U,. Ui • (H,"': (Hi,.,: If not, take bad balls of radii I, 3-i, U, 1,~, ~t ... ••• and a cluster point of their centers.) M. If (ZI, (s., ••• ... , s.) z.) E E- and W" ••. ,,.) , r.) E B-, then (S., 34. W., ... (sa, •.• , s., Z", '" fll, ... · · .,, ,.) r.) E E-..... Therefore if 8 and T are subsets lllbaets of E" E- and EE"+-. B- respectively, we may identify 8 X X T with a subset fi E-+-. Prove that if 8 and Tare subeet of T are nonempty, X T is bounded, or open, or closed, or compact, if and only if both 8 then 8 X and T are bounded, or open, or closed, or compact, reepective1y. respectively.

no......

U

35. Call. metric lIpaee apace ..-ei.", ......,.ei.", .,...,.. . ._ 35. eotIlpad H H ..,., ..._ . . . .". . _ . ....... _ ......... Pro .. daM • metrio apace illICpItIdIaIlJ . . . . . . . . . . . . . . •. eublequenee. Prove that " meuio .... II eequeatiaIlJ if ADd oeI7 ....,. iDtiDiM bdbdt.e _baa paint. ..,., eubIM ......... ... " poia'apace , ,.", . 38. Call. Call" metrie metric Ipaee . " , ........ H, for ..,., eYfq •• > 0, tbe \be ......... II . \be uaioa of • &aiM _t.e number of . . . . . . of ndiue ndiae .. Flowe . . ... ". . . . . . ..... UDioo ebed to !'loft .. II t.otall7 t.oWIy bounded Hand 0Dl7 only H..,.,......,. H..,., 1Iu. 0aueItr ~ . .. bouDded Hand .... Pro.. that the followiq oonditioae OD . . . . . . . . . . (cf 111. Prove followinc three OODditioae OR . .1Mtrio (cr. . . . . II, ..

.1."_,

.88) ) .... equivalent: equlvaleat: (i) 0) B is compact (ti) (U) B is eequentially compact complet.e (Iii) B is Ia totally bounded and cornp1ete. (Hiwl: That (i) implies impliel (6) tbe text. 'I'Mt ..,. (Hi*: (U) occun 0CCUl'I in iD the That (Ii) impIfII impJles (II) (iI) II II..,. foBo... from tbe That (iii) impliee (i) foUows the ........ ~t of tile the ... II-' pIOCIf plOof of 11.) ,I.) 38. Prove that an open opeD (cIoeed) IUbaa • metric . ..... eoDII88tecl.ud eubeet of "metric . . B II Ia eoueetecl if ADd 0Dl7 only wUoa of two DOII8IDpty (eIoIed) . ...... if it ill is not the diajoint diajoiDt UDioo aoumpty opal opeD (ebed) . . . . of ••

CHAPTER IV

Continuous Functions

Elementary analysis is largely concerned with realvalued functions of a real variable. Moet Most fil'8t first COUl'8es courses in calculus quote and use, but refrain from proving, such theorems about continuous real-valued functions on a closed interval in R as the attaining of a maximum and the intermediate value theorem. Among other things the present chapter proves a number of such fundamental facts on real-valued functions of a real variable. But it would not be reasonable to restrict ourOUfselves Belves to luch functions alone, for elementary calculus also &leo involves real-valued functions of more than one real variable, that is real-valued functions on subsets of lOme Euclidean space, apace, and it aleo a1ao involves finite sets of 88 for real-valued functions 01 of one or several variables, as some Euclidean example when a curve or surface in lOme apace is given parametrically or when complex-valued class of functions are considered. Thus the natural claee functions to consider would appear to be functions on one metric space with values in another metric space. lIuch functions that we define the notion of It is for such continuity, deriving from this generality the usual advantages of clarity of concept and obligation to do a thing only once. Our basic theorems will be general functions from one metric space results on continuous fUllctions to another, from which the basic results needed for elementary calculus can be read otT off by taking both spaces to be subsets of R. At the same time we metric spacee go forward, developing a number of useful concepts that uaually ueually do not appear in elementary COUl'8e8, COUI'8eB, such as uniform continuity. A final section on sequences of 88 functioDi will ilIuatrate concepti and will lunctioDl illust.rate the previous concepta provide us with further examples of metric spaces, useful in the sequel. This last section could more logically be placed at the beginning of Chapter VII than here, but ita its flavor is more nlore like that of the present chapter.

68

IV. CONTINUOUS FUNCTIONS I'UNC'I10NB

II. DEFINITION OF CONTINUITY. EXAMPLES.

By a function on a metric apace B E we of COUI'8e course mean a function on the let of pointe of B, and by a function with values valuel in a metric apace space 1£' E' we mean a& function with values in the set of pointe points of 1£'. E'. Thus if IiI J is a function E into the metric apace space 1£', E', written from the metric apace lpace B

J: E-+E', I: B-B', then to each point l' pEE /(p) E1£'. E E'. The function E B is il 8880Ciated 8IIOCiated a point 1(1')

J: E' will be called continuous at a point Po E E if, roughly speaking, I: EB -+ - 1£' points po are mapped Dlapped by I/ into pointe of E' that are near B that are near Po pointe of E /(fJo). Here is 1(".). il the precise definition:

Definition. E' be metric spaces, with distances denoted d and Dfdin.ition. Let E and B' d' reepectively, E' be a function, and let let". B. Then I/ is said respectively, let J: I: EB ---+ 1£' Po E E. contiftUOUB at Po ". if, given any real number If > 0, there exiIta to be continuous exi8tB &a real thatif1'E number I > Osuch thatifp E Band d(p, d(P, 'Po) Po) < a, I, thend'(j(p},/CPo» tbend'U
COUl'8e depends on I, 10 we could more accurately The number I of coune 1(.) instead of I. We Rick have written 1(1) stick to the notation a 3 rather than 1(.) ,(.) for notational simplicity, always bearing in mind that each (I. must have ite own I. its The definition may be reformulated by saying tbat that I is continuous at 'Po if, given any open ball in E' of center 1(".), po /(1'0), there exilte exists an open ball in E ". whose image under If is contained in the former ball. Another of center Po il that I/ is continuous at ". reformulation is Po if, given any open subset of E' contains/{fJe), there existe an open subset of B that contains!(Po), exists aD E tbat that contains 'Po Po whose image under If is contained in the former open subset. If Band 1£' subaete of R (so (80 that we have &a real-valued funcE' are both subsets Bays that IJ is continuous at 'Po 1', tion of a real variable) the original definition sa1l existe a I6 > 0 such that 1/(1'} if, given any fI> > 0, there exists I/(p) -/(",) - /(Pe) I < •f whenever l' E E and 11' - ".1 Pol < 3.I. This is illustrated in Figure 17.

I: E ---+ B' Definition. If E, E' are metric spaces and /: E' is a function, then contiftuOUl 01& if I is continuous J is said to be continuous on B E or, more briefly, continUOUB, continuoua, if! continuoua at all pointe of E.

11.

DUlIO'nOIl OOll'lllltJIT1' DIIftlll'l'lOIl or 01' 00IIft1l1ll,",

. ".

,r -/(;I) - /C;I)

-----•• ---------------

p.

PI

ExAMPL. The function I: R-R Biven by I(s) I(e) - .. EXAMPLIl 1. R - R liven II lor for . . tbM 1 II ,;s E R is continuous. To prove this \hie we have to Ibow abow that is OOIlti_ contiDuoua M U AIl7 •«> each Zo E R. We have to show Ihow tba~ for .., > 0 we can &ncl &nd • I> I >0 IUch tba~ tha~ I" whaJ.ever I,; Ie - _I < .. I. But But. Iince __ such I~ -:ell - St'l < • whenever I~ - :el1-I(e Zoll-I(,; +_)(,; I" + are)(e --->1 are) I -I (,; (e -- _- + are) 1 + "'(aI 2-.)(. -- -> I S (I,; (Ie - _I + 11_1)121_1)ls - _I. -I, we will have I" I~ -:ell < • if I,; &Del Ie I,; - _I 1. Hence we may take take'3 - min 11,

_I'

COIl"'"

EXAMPL. Let Ebe any metric apace, ,. • Sud EXAMPLE 2. Po. fixed point. point of B. TIleD the function I: E -. tl(P, JIt) Po> for B is - R given by 1(P) - "(p, lor all, all p E II II contiDuoul. show that 1 is continUOUl AIl7 Ii. To prove this we have to Ihow contiDUOUB at .., p. . poiU paiat "PI of B. But -/1-ltl(P, Po> - "(pa, tlCPa, JIt) Po> 1 tl(" H, Pa>, I/(P) -/
s

"(p,

if we have tl(p, 711) Pt) < • then a1eo -/(,.) 1 ThUi for . . JMIiD' poiU also II(P) -/(pa) I< .. TIl. MrI Pto correeponding to any •e > 0 we ean "" can chcae choole •I - .. The special epecial cue E - R, ,. lei •II ..~ Po - 0 Mon ehowe that ~ the fane&ion function 1.1

10

tinuoua. EXAMPLE constant function is i. coatift'" CORtinUOUl. In . this EXAMPLB 3. Any eonatant . .GIll . .•. andt.hepoint/(p) B' •.. the ......... , es....... and E' B' are arbitrary and thepoin~/(P)e Elf theaameforall, we .... ,. have tl t1(jCp),/(JIt) 0,10 any Po E Bud.., always (J(P),/ 0 we can take ato be a, _, poaitive poeitive DU1ftber. number.

,.e

e •.".

71 'Ie

IY. IV. COMftNUOua COllftNUOUli J1JMCDON8 nIIanON.

EnKPLB 4. metric space ia is continuoua. continuoUl. Elwm.B Owecanchooee'-e. .>Owecanchooee... The R Ihowt abowa tUt. that, the function s (the U8Ual usual way of '!'be IpICial . . .. .B- R writiDI nalline) continuous. writiq the identity function on the nil nne) is ia continuoua. all

Ex.um.. EuIm.II 5. s. The "step function" I: R .... - R liven by

Ie) s I(s)

{{o1

°

if:l:0

> 1·f S_ s_

.nee

°

is points, Binee I is CODltant when s:I: > 0 and ..., alao ia continuous continuoua at, at all DOnsero nonsero pointe, ia coaatant wIleD ss< < o. 1is noI at 0. o. For if we test continuity at the point 0. But But/ia tIOC coDWluOUI oontinUOUIat ,. positive lSI &hen we can PI - 0 with positive. S 1 then ean lind find no corresponding ,a > 0, since for any a and any z E (-',0) ( -I, 0) we have I/(z) -/(0)1-10 -/(0) I == 10 -11- 11 == 1. for&nJ' , > 0OandanysE EuIlPLB Ex.w....

6. The function I: R -.... R liven by

'0

is rational JlI if sz ia is not rational I(s) - { 0 if s ia

is eontaina both numberl numbers ia continuous at at. no point. For any &nJ' open ball in R containa that. are rational and Dumben numben that are not. that not (we already know we can find a rational Dumber. if II ill any fixed irrational number and number. in the baD, and if" ia &oJ' IUtlicientlyiarp intepr &hen N •a IUfliciently larp integer then the irrational number. number G + "IN will _ also be in the ball). ThUl for any" t S 1, a corresponding correeponding &oJ''' E R and any positive « a, > 0 cannot cannot. be found.

°

a

Eu...... 7. If I: ExAIIPLB J: B-B' B .... E' ia continUOUl and 8 ia is a aubepace aublpaee of B, H, Nlt.l'iction of I to 8 ia then the restriction is continuoua eontinuoua on 8. Thia ia ill clear from the defiDitioDa. definitioaa. followinl criterion for the continuit.y The followiDI continuity of a function from one metric apace into anot.her another it ia often uaefuI. Ipace Uleful.

.,1

,.......don. qaca and /: E --+ E' aII furu:Wm. function. TAen TAM rro,...cdon. 1Al 1M B, ll' B' 1M be "'*tC mtIric .,... _I: / " eonaiftuou for ..,., ",.,., Of"" open ",b,d U 01 E', 1M irwer" i""er.e i"..,. imGfl' lie ..,.,..... if iJ .." anti """ ii, iJ, lor

""*'

""*'

t-I(U) -- I, e B : 1(P) ,..I(U) IpEE: /(1') e E UI VI

ie _ . , . 01 B. u_OPM"o/B. thie, fint lirat. SUpp088 that I ia continuous. We have to show that. To prove thil, auppoee that/ill that if U C ,iaopen, B' _open, thenalsoJ-1(U) then -f"I(U) ia open. Let.,. U). Then/ ilopen. Let,. et-I( Ej-l(U). Then/ o. 0. Since Ia& ,. there is _ a ..a > 0 such that. ndiUl I is continuoua continuo.. at. that if ,p E 11 B p.) < < I,then < .. and "(P,,.) then tI'(J(,),/(p.) d' CJ(p), 1
_"(P,

°

e

11.

_nMlTlOR O. COII'nIiUlTY OONTlNWTY DUlIIITIOIi

71 TI

in the open ball in B of eenter center PI racliua • then /('1') Po and 1'Idi1ll I(p) is contained in the open ball in B' I(p.) and ndi1ll radi.. .. ., 10 that 1(1) 1(,) E U. That is, ii, ll' of center eenter I(pi) J-l(U) radius I. Since Po PI "11 wu J-I( U) contains containa the open ball in E B of center fit Po &DC\ radiua 1(U), the . &By let J-l(U) iI Conveftlely, auppoee IUppoee that for any point of t-I(U), . t-.(U) is open. Conveftle17, every eetj-l(U) _beet of B. We muat must Ihow show fN8l1 open U C B', the . . t-I(U) is an open aulat

u.

&ne\

r

ell',

that I is continuous point" E. For any e > 0 the I8t let continuoua at any point Po E B.

1-1«(open 1
ill Pt, hence containB contains the open ball in B is an open subset IUbeet of E B that contaioa contaiDl '" of center PI some radius o. Thus pEE Pe) < ',then a,then Po and lOme ndiua a • > O. Thua if P E B and d(p, JIt} d'(J(P),/(pe» meaos that I/ is continuoua continuous at '" Po, and thia this comtf(j(P),/(") < e. eo This meana pletes the proof. Corollary. 111 II/it tJ eontin1WU8 real-valued function on 1M the metric apace B E gO c:onJinUOUl real-fHllU«l/lmdion I1&m for an1/ """ lor _II (J II E R R

Ip E 8: B : 1(P) f(P> > at til care open . ..",." 01 B. 8. are . . 0/

tmd and

lit

Ip EE 8E : ICp) /(p) < at tit

lit

For the Beta aetI I~ ~ > aJ and Iz (~E z < til are open open lubeeta 8ubsets Iz E R : z> ER :z ofR.

following result rault is uaually continuous function of The foUowing usually parap1uued paraphrued "a " a continuoua continuoua function is a continuoua a continuous continuous function".

Propoaldora. Let, Let H, 8, 1£', E" be metric apacea, I: Propoa'don. E', 8" J: 8-8', E-+B', ,: E' -E" ..... 8" tmd , care g 1M /unt:lior& ,,oJ: 0 I: E if /I tIfId are c:ontinuoua, eontinuoua, 10 ia tAB Junction -+ E". ~, if PI Po E E and tmd IJ g More ",eciNlJI, ia continuoua eontinuoua at Po tmd tmtl , g u eonCinuoua continuoua at Jl(pe) (pe) E E', """ I1&m , 0 /I it Pee g continuous c:onlinuoua at p.. . Jun.cliqu. TAm JUftditml.

We need only prove the latter, more preciae, precise, part. Let d, Il, tf, tl, tf' ti' denote metrica. Suppose et > 0 i. the three metrics. is given. Then, linee Bince , iB is continuOUl continuOUI at l(pe), there exists exiata I• > 0 luch that if 1q E ll' /(,.), E' and d'(q,/ 0 auch E B and d(p, JIt} thiI such that if P pEE Pe) < " tf(jCp),/(pe» < •. then tl'(J(P),/
<.

'0

arpment iB The above argument is illuatrated illustrated in Figure 18 on the next pap. page. verBion of the lut The weak version last propoeition, proposition, where I: /: B -..... B' E' and - E" ll" are UlUmed &IIUIIUId to be continuoua ,: E' .. continuoUl everywhere, can be proved very previoua criterion for continuity, as followa. limply by meana of the previous follows. U If U is any open subset IUt.et of ll", ill B", the continuity of, of , impliea implies that rl(U) U) iB is an open IUbaet of B', 10 8Ubeet 80 the continuity of of/limpliea implies thatt-I(,-I(U» that J I('-l( U» iB is an open wt.et subset E t-I(r'( U» if and only if of U. But for any U C 1£", 8", if p E B then p EJ-I(,-l(U»

,-1(

11 'II

IV. CONTINUOU8 CON'J'ltf110tJ8 FUNCrlONB rtlNCTlON8

f(p) E g-1(U), if and only if g(J(p» g(J(p» E Thusf-l(g-l(U» .. == /(p) g-I(U), which is true tmeif E U. Thusf"I(g-I(U» Cs1 therefore shown that if U is an open subset of E", (g 0 J)-J(U). f)-I(U). We have thereCore I( U) is an open subset oC then (g 0n-I(U) of E, which implies that g 0/ is continuous.

n-

B

--------------------~ ,-I Flouu 18. A continuoU8 continuous function is continuolIL continuous. FlG11IIII continuoUi function of a eontinuoUi

12. CONTINUITY AND LIMITS. Consider the following question: Let E, E' be metric spaces, let p. E E, H, let elPoI elp.1 be the complement of 'p.1 po IPoI in E, and let /: elp.lefPo) -+ E' be a function. Is it poasible possible to extend the definition of /J to all of E in such lueh a way as to obtain &a function from E into E' that is continuous at 'Po? p.? & case the answer to this question is trivial, and that is the case In one ease cue p. is not a cluster point of E. For if p. where Po Po is not a cluster point of E then _II function from E into E' is continuous at po. p.. This is 10 Gfty 80 since there exists ball in E of center Po p. that contains only a finite number of points a baD pointe of H, E, p. and smaller radius that contains only the point flo. hence &a ball of center Po anlaller radiul Po. pEE If ,a is the radius 01 of the latter ball then the statements that "E E and p.) < ,a imply that p - ".. d(p, 'Po) Po- Thus for any /: J: B -..... B' and any any p fJ E B that d(p, Po) Po> < a we have d,'U
D"Ifinltiora. Let E, E' be metric spaces, let Po Definition. po be a cluster point of E, e(p.ltil and let/: elp.J ..... E' be a function. A point q E E' is called a limit oj /fat Po if the function from Einto E into E' which is the same as/on asf on elPoI e(,.) and which takes on the value q at Po is continuous at Po. po.

n is useful to reword this, without directly using the notion of conIt tinuity.

12. ,2.

CON"nNUlTT AND LlMlTII LIMITS CON1'1NUl'I'1'

73 1a

Definition'. Let E, E' be metric Sp&C5, spaces, let Po be a cluster point of E, and let/: 0/II at ". Po if, let I: elPoJ e 1Pol --+ - E' be a function. A point q E E' is called a limit 01 given any E > 0, there exists a 615 > 0 such that .if jf pEE, P""" Pt, and d{p, d(p, Po) < 6, 15, then d'(f(p), d'(j(p), q) < f.E.

p"

Given E, E', a cluster point Po po of E and I: f: eIPoI-E', e(Po) ..... E', there can aiR exist at most one limit of I/ at Popo- The argument iB is the &arne same aa 88 that for the uniqueness tI E E E' are two uniqueneaa of the linlit limit of a convergent sequence: If q, q' limits of I/ at po, then given any Ef > 0 there exists a IJa> 0 Buch such that if pEE, p ",. po and d(p, Pe) Po) < 156 then tl(j(p), d'(/(P), q) < •t and l(j(p), t1(j{p), q') () < .. t. Since po luch that Po is a cluster point of E, there actually exist points pEE such p ~ ~, so 2•. P ;po! po Po and d(p, Po) < 15, 80 that we can deduce that tl'(q, q') < 2 •. Since this is true for any Ef > 0 we have tl'(q, d'(q, q') t') --= 0, 10 tl -.. q'. tI. 80 that q Po exists, exiete, then since sinee the If, under the above conditions, a limit of I/ at ". limit is unique we nlay tAs limit of I at ". Pe and we denote this tIlii may speak of the

.

/(p).. lim Iim/(p)

...." "..

The statement that Iim/(p) lim/(p) exists implies that we have metric spaces 8p&ce1 "..Jto .... "

E and E' in mind, that po Po is a cluster point of E, and that we have a function f: elpo) Po- In I: e 1Po) --+ - E' such that for some lOme point q E E E' q is a limit of II at ".. discussing bappena discu88ing limf(p) lim/(p) we may be explicitly given a function IJ that happens ....I'e "..".

to be defined at Po, po, but this is immaterial: the limit of I at ". ,. doee does not depend on whether or not fI is defined at Po po nor, if it is, ii, on what its ita value Po is, but rather on the values of I{p) I(p) for p near, but distinct from, ".. at po Pt. In the above work we started with a function I/ which waa was de6ned defined at metric space but one, but it is poasible all points of a Inetric possible to diacuu diICUII limits timite of functions whi(~h are defined on relatively small sinall subsets of a metric IJ)&C8. apace. metric spaces and that we have an arbitrary For example suppose E, E' are Inetric haa at least leaat one cluster point in B, subset SeE that has E, topther together with a /: 8 ..... E'. If p. function J: po E E i. is a cluster point of 8 we can conaider eouider Iim/(p) relative rfliatilJfllo IPoI the lim/(p) to Ihfl the BUbapact subspace au SU 11' 01 01 oj E and one then speab speaks of of"",

a. . .

a

"..'"

....I'e

a.

limit ojJ{p) oll(p) as aa P approachu Po on 8. Thus a specific metric space E m1llt p approache8 mUit be bome in mind in considerinllim considering lim I(p), borne /(p), and in the lut cue the space epace to be

..... ...."

bome in mind is actually not E, but the subspace 8U borne 8V 1".1. (Pel. The moat most B that frequently arising case iB is that in which If is defined on a part of E includes all points of an open ball in E of or center "., Pe, with the possible Po- Here we maintain the same notation exception of Po. liml(p) lim/(p)

"..'"

....I'e

I may without any reference to the fact that that! Inay not be defined far away from fronl Po: it is enough that I be defined near Po, except poesibly po; possibly at ". ,. itself. it8elf.

7.

&Y. ClOIft'lIfVOUIl

""emo...

w. pve pn our fint, &nt. definition of the limit Iimi~ of & We a function in terma tenna of cooc0nde&ne continuity in tenna terma of limite tinuity, but one can equally well define limits of functioaI. If If I: B - II' _ a& function from hom one metric IpIee functiona. B' ia apace into another and Jtt E B, ia continuous aDd PI E, then II_ continU0U8 at &~ Po " if and only if, if Po " ia _ a& cluster cluater point of B, tbeo tbeD 01 -/CPt)· ...lim/(P) ,.. -/
-

limp" limp. - q 9

if and only if, for the subapace 1,~,~,~, IUbapace 10, 10. 1, ~. U.).( • ... )J of R and the l(l/ft) - p. aU fratfunction I: 11, ~, U, H, ... )J - EB liven by 1(l/ra) p" for all 1,2, a, ..., ... , we bave have lim/(P) - 9q..

.....

Tbia ~~ ia _ euy to verify. Aa AI& a& matter of fact the notion of continuity TbiI.tatement a function) ahould be reprded .. (or the equivalent equivaleD~ notion of limit of & fundamental, with converpnt eequencee u. ..rul devicel, in IIeqUeDCeI . uaeful technical devicel. lpite of their earlier earIi.. introduction in thia ted. have been more mon epite text. It would bave ·lnaturll". althouch althoqh a& bit bi~ lonpr, loDpr. to tint fiIR prove theorema about continuity ''Datural'', (or limitll NIU1te on limite of eequenceL limita of 01 1unctioM) fuDctioDI) and then the analogous ~ a.ulteon lequencel. of our ichu ideu .... IoIPcalI7 inteldepeadeDt. aDd the followiDa Of ..... COUIIe aU 01 are loPcaII,y inteldepeodent, aiuI followinl pIOpOIltloa, Ullful criterioD of continuity, could ...., a1Io han bave propoIltlon, wbich wbleh .... a UIIful oriterioD 01 be.- . . . to lin pve aD altenate cIeftDhioD continUOUl function (and beaee hence bem .... &D alterDate cIeftDitioD of continUOUI of the limit a function) conv....' IeqUeDCeL IeqUeD.ceL 6mi~ of oIa fUDCtlon) in __ terme of coovera-~

p.,..

rr.p.......

lAC H, E, B' .. ....eric tJUIric . .".... TAcrn G/tI.'Itt'Jitm • fwt,ditm I: BII' .. ...........doa. lM . . . TINa B-B' .......... ., PI iI,for..." poi"" Pl,",,,, , •• ,., Pa•••• ........ Jtt E B iI -• .." onl, ii, lor ..., ~ OJ poinu ...

• B. "'B .. . .., 1Aae

-

limp" limp. - fie PI

lim I(p.) -/(p.). lim/(p,,) -/CPt).

8appOIe fint &nt that I ia _ continuous continUOUI at &~ PI •• ,.. 8uppoee fie and that ,Pl, ,., Pa Pa,•••• • •• iI ia a 1IIlUpaintilin E that converpl conv_ to ".. ,.. We have . . . . . of pointe in H bave to abow Ihow that tbat the ....-/~,/(H./(p.) ... conftl'ltll conv.p8 to Ic,.). I(p.). Oiveo ....... 1<-N,/fH,/<Pa),• ..• Given •41 > 0, the con-

&.

.....,. 011_ PlhnpIlt. that there tb.-e it iI a a > 0 such Rch that . .011 at. '" impIiea that. tbat "(j(P),/(p.) d' (J(P), ICPt» < •

13. ,3.

COII'ftIiU1TY O. 1IA'ft01lAL IlAftOIl... 01'''''11011. OPDA'IIONa OOIl'ftIlUl'l'Y 0.

11 71

whenever pEE and d(p, PI) < a. Since Binee Pl, PI, Pt. PI, Pa, .• .. •• converps converseB to PI there is Pe) < a I for all 1& > N. Hence if ia a positive integer N such wch that d(p., PI) A shOWI that 1(pi),/c'w,/<Pa), l,J<Pa), ..• •.• 1& > N then d'U, then I is continuous at JIo. P0I (ps) , I C,W, I(p,), .•• ... of E' converges to I(p.), We do this is that I ia is not continuous at thia by lupposing supposing the contrary, that ia Po, pointe PI, and deriving from this assumption 888umption the existence of a sequence of points PIt 'PIt Pa, PI, ••• ••. converainl convcqing to PI wch Pl, Pt, Buch that 1(p1),/c,t),/<Pa), J,J,J<Pa), ••• doee doeI not converge to ! 0 such that for no number a> 0 is ia it true that whenever pEE and d(P, necessarily d'U(P)'/(po» tlU, 1<Pa), sequence pa, .• converges to JIo. po. However ICPl), l
a

,3. THE CONTINUITY OF RATIONAL OPERATIONS. 13. E-. FUNCfI()NS VALVES FUNCflONS WITH V ALVES IN 1:-.

functiona on a metric space E (or indeed on any set) can Real-valued functions be combined in the usual way by the rational operationa of addition, 8UbBUbThUB if I, g traction, multiplication, and division. Thus 9 are real-valued functioDB functions g, I - g, Ig and Ilg, on E, we have the real-valued functiona functions IJ + 1,1JIg, given by

+

(J + ,)(P) g)(P) -=-1(P) 1(P) ,(P) g(p) (J (f - ,)(,) g)(P) -/(,) - 1(P) - ,(P) g(p) (J,)(:P) :II J(P),(P) (fg)(p) - 1(P),Cp)

(~)(p) =- ~~~ p E Ej in the lut for any 11 last cue II, ia is of course coune not defined at any point p ,Cp) so such that g(p) == 0, 80 thatllg that/I, is a function on (p E E: E : ,(P) g(p) 7' '7" O). 0).

PropoaitiDn. Let Lee I and g be real-tH&lucd Propoaition. NOl-fNllued /UftdUrn.. J'Ufldiona on Ga metric IJHIU apace E. 11 II /I and , are contmuoua Po E E, tIaen then 10 80 are the ItmCtion. l'UfldUm8 1+ "g, c:ontmuoua at a point poi'" PI laat under the proviao tAat II-"Ig - " I' and JIlg, I" the lGat under tAe tI&Gt ,(p.) ,(PO) .. 0 (in tllAim which CCI8e cau g(p) ~ pi 0 1M lor aU poWa poiftta P p in ,(P) '1ft aome open ball 01 center Po}. Po). Thia can be proved directly, but it is easier to deduce it from work This already done. By the lut proposition, to prove that I 9 is continuous at .ufIicee to .how show that if PI, PI, PI, .. is PI it suffices Pa, .••• i8 a sequence of points of E that conVel'leB to 'Pe, Per then the sequence 1(Pt> 1(Ps) ,(Ps), I CPt> + ,(pi),J<Pa) converges g(Pl),/ g(P.,),J<Pa) +

+,

+

+

+

76

IV. OON'I'INtTOt18 OON'l'lN1701Jll rt1NCTlON8 ruNCTIONII

,(pa), . .. converges to II(Po) (Po) + ,,/(p,) I(po) and ,(P.) ,(pa) reapectively, respectively, 110 I(pi) + ,(p1)./ ,C",),I<Pa) g(Pa), ••.. • • to I (Po) + ,
+

+

+

+

+

Ig(P) ,
......

...... ......

"'Itt

,(p» - lim/(p) lim I(p) + lim ,(P) ...... + ,(P» ............ ... ,<1') lim (f(p) - ,(P» g(fl» --= lim I(p) - lim ,(p) ...... ............ ,(,,) Iim/(p),(p) lim 1(1') 1(P) • lim ,(P) lim ......l(p)g(1') - ...... ...... g(,)

lim (f(p) ~...

~...~

~~

~~

~~

a:

~~

i/lim and, if lim ,(p) g(p)

......

po! ~

~~~

0,

"..".

......

limj(p) lim/(p) lim I(p) /(1') .. ~'" ,(1')• lim ,(p)" ~..... ,(p) ".Jte g(1')

...... .......

coro1lary means, means. fil'Bt, The corollary firat, that the left-hand limite limits exist exilt and, second. 8econd, that they are given pven by the appropriate formulu. formolu. It is an immediate COllIeconsethat; quence of the proposition and the fintt first definition of limit of a function.

Lemma. For eacA 1, 2, ...• ... , ". ft, 1M the fundion function :1:0: Xi: E" E- --+ R dejinetl tkjined by eac.\ i == ... 1.2, ••• eontiftuoua. • •• , a..» - CIi i. " continuoua.

z.( :I:o((ai, (tilt Ot, CIt.

a.» ..

We have to prove that :.:, %i is continuous at any given point Po po E E-. Ea. For any pEEpEE" we have p = == (:':I(P), (:1:1("), s,(P), %1(,,), ...• •.. , :.:.(P». x..(p». The inequality 1:I:o(P) -z,(,o)1 -~) 1~ V(ZI(P) V (:':I(P) -Zt(po»I+ - :':I(Po»' ... + (x,,(p) (z.. (p) --z,,(Po»t :.:.. (".».... P.) Iz,{p) == dept d(p,po) d(P. Pe) Po) < t!t then also 1:I:o(P) < t!.t. Thus :.:, shows that if d(p, IZi(P) - z,(Po) %i(Po) 11< Zi is conflo. tinuous at Po.

+ ... +

'3. 13.

CON"I1NUlTY 0 .. .... 1U.'I10NAL OP.aAftON8 CON'11I1UlT'f or 110IlAL 0 ......11011.

n

The previous proposition can be combined with the lemma to live many examples of continuous real-valued functions on Ea. If .e we recall that any constant function proposition refunetion is continuous and apply the propoeition peatedly, we see that any polynomial in ZI, ZI, Zt, %I, ••• , z %" coefficientl in .. with ooefticienta R is continuous. (For example, ZI' Z11 - Za Zt + ~ Zt %1 - n-. ~ ia is .. c0ncontinuous function on E'.) Furthermore any rational funetion function ((a .. ratioD&l function is the quotient of two polynomial functions) functiona) ia is continuoua contiououa whenv wherever .. the denominator is not zero. (For example %1Zt!(%,1 ZlI) ia is .. a continUGUa continuoua ZaZt/(ZI' zI) function on J.:2 E' - f/ (0, 0) I.) a function, then the imap iJ11&l8 in .lI-. If E is a metric space and J: f: E ---+ E" .. of any point I' pEE E E is the point

+

+

f(p) j(p)

= (ZI(f(p», Zt(f(P», ... == (Zl(J(P», Xt(J(P», •.. , z.(f(P»). z.(J(P»).

Thus IJ is determined by the n component 2:1 0 0 I, Zt %I 0/, 0 / , ..• ••• ,, z. z,. 0 0 I. componen.l funcl,ioM ~ z. ConverBely, metric 8p6C8 space defines defin. Convenely, an n-tuple of real-valued functions on a metrie a function E". The foUowinc following propoeition IumdleI bancI1eI funetion from the metric space into Ea. all relevant continuity questions. Proposition. IJXIU t.IftIl'/: tmtt/: E -..... Ea E" tJG ~ ft,mdioa. 1'_/ TAen Iii Proposition.. Let L« E be atJ metric IfNI" .. continuOUI tJI, a tmd only if eacA eac1& cmnpcmene comJ'lOl"ft' /f,mI:JiIM /uftd,iM& .. s, 0 It continuoua tJt tJ point po Po E E if il mad z .. 0 IJ i, iI continuoul continuou. tJt p.. ... , z" tJI, PoThe "only if" part of the proposition comes trivially from the lemma, for if J is continuous at Po then 80 is the continuous function of 01. eonti_ if' .. conti .... Zi 0 J. , . To prove the "if" part, suppoee function Xi suppose Z. %1 0 I, ... ...•, z. %,. 0/ 0 I eontinuoua con~D1I01I8 0 f)(p)-%I(/(P»,(Zs o nc,)e> O. Noting that (z. - ZIC/(P», e,..oJ)CIJ) at Po and let e>O. (Z1 oJ)(p) %t(J(P», a. > 0 lUCIa XI(J(P», etc., for each ii = 1, ... , n we can find a It such IbM t.bM

!%i(J{p» < IXi(J(P» - %tC/(Po»! Xi(/(P.» 1<

In

whenever I' pEE Ii, d denotinc denoting the diatanee distance in B. E. If we .let E E and d(p, Po) < 8;, = min la., /a., ... , a"l, then if I'pEE E E and tl(p, 8a == d(P, PI) Po) < •a we have

!Zi(J(p» Ix,(j(p» - Zi(J(p.}) x,(j
80

In

that

v(Z.(f(p» ZI(J(Po»'1 + ... ..(j(po»'1 V(ZI (f(p» - %l(j(Po»)i ··· + (z.(f(P» (z,.(f(P» - z 2:.(/(1'0»)1

< .J(JnY <.J{Jn Y++...... +(J-;ar-~ +(J,aJ'-~

This proves I/ continuous at p.. Po-

lemma becomes a special case of the "only if" put Note that the lemnla part of when we apply the latter to the identity function on Ea, the proposition ,vhen which is known to be continuous.

TI

lY.

OOII'IIIIVoua I'UIIG'ftOIl8

14. CONTINUOUS FUNCfIONS nJNCTIONS ON A COMPACf COMPACT METRIC SPACE. TIaeorem. 1Al Ld B, B' be tneIric f: E -+ E' metric 1JIGt'U, 1fItJCU, I: B-B' cmnpact, 10 eo;' ;:",age/(8). TAM il B;, B if compad, iI ita image /(E). We must /(8) muat show that if 1(8)

I:: 0::

(J G

contiftU0U8 function. etmlinuoua ,"MUm.

1/(,) (f(P) : p l' E 81 BI is contained in the union

of •a eoUect.ion collection of open sublets subsets of E' B' then it is contained in the union of a of

finite number of these open sublets. f U,liEI is a collecsuheete. So 80 IUppoee 8Uppose that (U,I'EI tion of open IUbeet8 8Ub8ete of E' B' whose whOle union contaiDBf(E). containe/(8). Since/ia Since I is continuous, continuoua, eMIl invene inverse imapJ-I(U.) imapt-I(U,) iI is an open sublet subset of 8, B, by the &rat first proposition each of AIIo, for any ,l' E B we have 1(P) I
',1.1.

Since B Ie is compact Ulere there is a finite sublet subset J C 1lauch 8uch that B V t-l(U;). E .... -= U j-1(U i ). IEJ iEJ

I(B) .. - V 1(j-l(U;» U;.i • Thu8/(8) Thus/(B) is compact. Therefore I(E) f(J-'(U.» C V U iEJ

fEJ -=J

This theorem bas baa two extremely important immediate cODBequences. consequences. To atate the int, fint, let 118 ua .y eay that a function I: E B .... - E' B' from one metric apace into another ill ie bountUd image /(8) I(B) is bounded. In the special ...-ce bounckd if the imap . . that 1 I ill ilia cue a .-I-valued real-valued function on B this meana meane limply simply that there is a number MER lOch such that I/(P) I :S M for all l' Dumber p E B.

Corollary 1. E' Ga conIiftuoua function. I. Let, 1Al E, B, B' be tneIric metric 1JIGt'U, 1fItJCU, I: E B .... - B' etmlinuoua/unclUm. il B ;, compact, /I;' TAM if if cmnpad, .. 6ot.mtW. bounckd.

subset of a metric space apace is bounded. In particular Reuon: any compact subeet Ule compact sublet i8 bounded. the auheet J(E) I(B) of E' B' ill lut result reeult is false falee if compactne18 compactne88 is omitted; omittedj for example, consider coneider The Jut the function I(z) f(z) - lIs l/z on the open interval (0, 1). If I1 is a real-valued function on a metric space apace E and PI Po E E we .y say that II attaim fJIIGin.Ia CIt PI f(Po) ~ 1(1') J(p) for all l' pEE, a mazi"*,,, mazimum at Po if I(Po) E E, and we say eay that II GUtJina pEE. oUGin.a a minimum at ,. Po if /(",) I(Po) :s; S /(p) /(1') for all l' E E. C'..orolla,.,. Junction Oft on a nonempt1l compad Corollary J. A contiftuoua etmlmuoua retJl-tJGlued real«llued lunction compact metric aptM:e ~ attGina ma.zimum at lOme point, and also attain. oUGi", a mazimum al eome altaina a minimum ., al aome point. For let E B be •a compact metric apace, I: /: E --+ -+ R a continuous function. Then J(B) H, hence closed and bounded. If E is 'l1len I(B) is a compact sublet aubset of R, ia

,2

ill 1(8), and the 1ut then 10 ie/(/l), ~ pIOpOIition pmpoeition of '2 of the preceding us that I(E) hal has a P'e&tc.t greateat element, and &lao &leo a leut least element. chapter teUa 118 If U ". Po E E Hie E ill choeen 10 that 1 ICPt) 18 ill the P'e&teat greatest (leut) (least) element of /(8), I(E), then

nonempty DODempty

I attaina 1 attains a maximum (minimum) at ,..

Corollary 2 is falae if the compactneIB eyen for com~ condition is omitted, ellen functiona; for example, consider conaider the function I(s) I(~) .. bounded functions; - s~ on the open interval (0, 1).

FloUD 19. A contiDuoU8 nal-valuecllunctioD OIl a c10Ied cIoeed interval FlQUBIl oontiDU0U8 real-valued luactioD GIl attainI a maximum and a minimum. in R aUainl

If E, E' E - E' is a continuoua continuous function then U H, B' are metric IIp&eeI IJ*l'I and I: B B' 18 E B and any •e > 0 there exiata existll a real number •3 > 0 IUch such that if ppEE E H and d(p, tI(p, pi) pO) < •3 then t1 d'(j(P),/ > 0, then Ia will depend on the point PoPo. A. As Po varis variee 10 in general will a, and it mayor may not be true that we can find a •a that works simultaneously for all ,.. If it happens happena that we can find such a I, a, worb and can do this for each •e > 0, the function 1 I will have lOme some especially nice properties. This leads to the following definition.

p~ liven lAY any ". Po E

80

IV. CONTINUOUS FUNCrtON8 FUNCTIONS

DpJinit.ion. 8P&(1IC8, with distances denoted by Df'jinitilJn. I~t Let E and E' he metric spal'CS, E' be a function. Then IJ is said to be d and d' respectively, and let I: E -+ A" unilormly continuous if, given any real number Ef > 0, there exists a real uniformly d'(j{P),/(q» < E. f. number 8a> > 0 such that if p, q E E and d(p, q) < athen d'(/(p),!(q»

If it happens that a function J: I: E -+ E' is such that for a certain 8ubset subset S S of E the restriction of If to S 8 is uniformly continuous, we say that J is llniJortnly unilormly continuous on S. 1"hu8 Thus uniform continuity on E is the same thing as unifonn continuity. function/: --+ E' is continuous: It is clear that a uniformly continuous function I: E -+ to check continuity at a point Po po E E just set q = po in the definition. The = Po next theorem will state that conversely if f/ is continuous then fJ is actually uniformly continuous, provided E is compact. If E is not compact then continuity does not in general imply uniform continuity. Here are two functions on the open interval (0, 1) examples of continuous real-valued functione uniformly continuous: that are not unifonnly

(1) The function I/ given by J(x) I(z) = l/z l/x for all x E (0, 1) is continuous but not uniformly continuous. Continuity is known. Uniform cona > 0 we can tinuity is disproved by showing that for any Ef > 0 and any 3 find p, q E (0, 1) such that Ip - ql < 8a and 11/p - l/ql > E. Specific such p, q can be found, for example, by taking q == = p/2 so that the eonditions become bccolue p/2 < a, l/p > E, the pair of which will be satisfied if 0< min (26, lie, l/f, 11. o < p < Inin . (2) The function fI given by I(x) == sin (l/x) (l/z) for all x E (0, 1) is cont.inuou8 cont.inuous but not uniformly continuous. To check this example ,ve we assume (these will be that the easier properties of the sine function are known (theJe I is continuous, and moreover rederived anyway in Chapter VII). Then 1 (l/z) II::::; ~ 1 for all x E (0, 1) any 3 since Isin (l/x) 6 at all will work if Ef > 2. But I, no a a. if ef < 1, 6 will work. For suppose that 0 < Ef < 1 and that 0 < 8. If we then take n a sufficiently large positive integer and set p = 1/(2111&), == 1/(2rn), r/2), IJ{P) - I(q) f. q .. =- 1/(2rn + ... /2), we get both Ip - ql < 36 and I/(P) f(q) 1- 1 > e. Theorem. Let E Gnd and E' be metric apace. apace' and J: E --+ -+ E' (I a continuoua function. 11 E gill compact, then J iI Junction. ill uniformly uniJormly continuoua. continuous.

IItt will be instrnctive'to instr1lctive to give two proofs of this theorenl. theorem. In each e8.(~h proof with a real number Ef > 0 and try to find a number 8a > 0 such that we start ,vith if p, q E FJ 6 then d'(!(p),/(q» E are any points such that d(p, q) < a d'(j(p),J(q» < E. E. ~'or For the first proof we find, for each pEE, a number 6(p) a(p) > 0 such a(p) then d'(/{p),!(q» d'(j(p),/(q» < e/2; f/2; this is possible that if q E E and d(p, q) < 8(p) since f is (('ontinuous ontinuous at p. Let B(p) be the open ball in E of center p and 8(p), with p ranging over all radius 6(p)/2. tJ(p)/2. E is the union of the open sets B(p), ll

14.

"UNCTIONS ON A A COIIPACT COMPACT ...... ...,..0 &PAOlI PUJfC'l'lONB 0 BP.u.

11

the points pointe of 1£. B. Since E is compact, it is the union of a finite number of these open sets. Thu8 .y PI, PIt PI, Thus there exist a finite number of points of E, I&J .•. ,p", , p.. , such that E = ... B(P1) B(PI) V B(Pt) V·· V ...· V B(p.). Now define . ... min (B(PI) /2, '(PI)/2, /2 J• We claim that this aI I&tiefieI sat" our (6(pM2, 6(Pt)/2, ... , 8(p.) a(p.)/2). demands. For suppose that p, q E E, with d(p, q) < •. 8. For lOme i-I, i - 1, 2, ... , n we have p E B(p.), B(p,), 80 that d(p" p) < .(,,)/2. AIeo d(p" "(p" ,) ~ '(p,)/2. Alto dept, a(Pi). Thus d(p" dCPi, p), d<'Pl, d~, I) 9) < '(p,). 1(pJ. d(p" p) + d(p, q) < 6(p.)/2 6(p,)/2 + a ~ a(p;). I~y 8(1''') W88 WR8 c~h08Cn (~h08Cn we have tl(J(p,),/
+ :s

+

d'(j(p),!(q» d'(J(p),!('l» 5: :S d'(j(P).!(Pi» d'{j(p).!(p,»

+ tl(J(p,),/(,J» d'(J(p,),!('1» < i~ + i~ - . •

and the first fint proof is complete. For the second proof we use U8C the nlethod, _mini UlUn1ini t.hM that for tho indirect method, our given number ef. > 0 there is i8 no •I > 0 IUch that if ,. p" I E B and d(p,q) fl, and we derive a contndiction. contradiction. 87 B7 d(p, 'l) < a 6 then d'(/(p),!(q» d'(/(p),/('l» < t, assumption, 2, 3, ... the number nUDlber lIn l/n is iI not a• auit.able auitable &88umption, for each n =- 1, 1,2,3, candidate for a, 80 that there exists a pair of points ,p., E B auch that •• fa E d(p", fl. Thus we have a aequence eequenee of 01 d(p.. , 'lqra) .. ) < l/n lin and d'(!(p,,),!(q,,» d'(J(p,,),/(q..» ~ t. ordered pairs pail'8 of points (PI, 91), 'll), (PI, /]1), (Pt, ••• wit.h (1'2, Q2), (Pa, /]1), 91), ..• with the propertieI p ~ t.hM that Jim d(p", d(p.. , q,,) q.) ... ..» ~ f.f for all n. Since E is compact, the lim = 0 and d'{j(p..).!(q d'(!(p,,), /(q,.»

e

. ... ~

p., f)2, '/h, Pat pa, . .. has h88 a convergent subsequence. Henee sequence PI, Hence we may repla('c (Pl. 'l.), ql), (Pt. (1'2, q,,), (Pa, 91), .•. . .. by a subeequenee subsequence balDCh in such rcphu'c the SeqUCIU'C seqUCl\('C (Pit /]1), (Pt, manner that we luay may 8S8UlllC &88umc that the sequence acquence PI, 'PI, a nlanner Pt, Pa, ••• eonverp. CORverpe B, st.ill still luaintaininl maintaining the to a point Po E J:, thc conditions liln = 0, lim d(p.. d(p"., q.. q,,)) ... }4'ronl From

(/(1',,), /(9.» ~ e.to d' (J(P.).!(,.»

the inequalities d(,., 1'..) P.) + d(p., o :S ~ d(q., d(q", Po) :S ~ d(q., d(p.., 'PI) Po)

and the equations

--

lim (d(,., (d(q., P,.) p.) Jim " ..m

+ d(p.., d(p., ".») 'Pe» -

:II:

.-

-lim d(q.., d(I., ,p..) linl ..)

+ lim d(p., tl(p., Jtt) Po> --

0

..........

lim d(q., 'Po) •• aIeo it follows that Iinl Po) - 0, 80 that the aequenee sequence ft. 11. fit fI, fI, fa, •••• &I" .

.~

converges to 1Jo. 'Po- Thus the continuity of I/ at Pe 'PI implies that convergM tha' lim/(p.) .. lim/(I.) == Iitn/
,,-.cD ---

.... .....

For n sufficiently large we therefore have For" d'(j(P.),/(Po» d'(j(p,,),!(Po»

< ;, tl(J(q.),/CPo» d'(J('1"),!
implying that d'{j(P.).!(q..» 5: :S d'(j(p,,),!,/('1"»

< ; + ; - ..to

d'(J(p.),!(q.» ~ ~ ·f. contradicting d'U(p,,),!(q,.» "e. This ends ~ ~be second proof.

••

..,. OOlftllnJ01JII OON'lPfUOUli I'VIfC'ftOD J'VIfCI'IONa IY.

I S. So CONTINUOUS J"UNCI'IONS nJNCI'IONS ON A CONNECI'ED METRIC SPACE.

Tlaearem. 1M II' ". lie mdric meCric tpGCeI, .,.,., I: E-E' B - II' 0a continuoualunctitm. continuoua lunction. TlaeDrem. L.e E, E' T_ if B if connedtd, 10 ao it if ill imtJ(Ie I(E). TAM" it~, ilI . ./(E). To prove plOve tbiI ihie we may without loea 1081 of generality generalityaaaume uaume that E' -/(E). -/(B). W.ehallUlUDle W. aba11aaauIDe that B' E' -/(E) iI is not connected and derive a contradiction. Since I(B) ilnot is not connected we can write I(E) -... A V B, where A and B Bi. . I(E) an diajoint oonempty open au'" subeeta of I(E). By the &rat firat propoeition proposition of II, t I, are . each . of the theaet.a/""I(A),JI(B) ""f"I(A), f"1(B) iI ill an open subeet subset of E. We therefore have the expl...wn of E .. as G,.,.Hm /""I(A) V f"1(B), /""I(B) , E - .r-a(A)

disjoint nonempty open subseta. This contradicta the faot fact the union of two diljoint ill 00IIIleCted, connected, proviDe the theorem. that B is .

,I

'I 'I I I I I

I I .1 I I I I

·• It It It

II

-1--II II I I

, ,, , I I I II

6

Corollary (lnr. (In,.,,mecflae. t1aeoNm). 11 a, b E a, R, Ga < h, b, And c:orou.,,. ...... tIaeoNm). 110, _ II it • eonIifttIouI ......lu .......,.,., ~ /tIttt:Wm em 1M'" lite clotetl intflnIcIl ifttmtGl (G, [a, bl, b), then lor can,"" anti real ........... ...... " ""'/(a) _/(b) fIwe t1we . __ (a, b) . I'IIda . . . . . ,. . . . . /(0) -/(b) . . CIt.least one point c E (G, . "/(c) IAaC I(c) -,.. - ". since (a, 10, bl b) is connected, 10 eo is I«(a, I([a, bD. The (almost trivial) &rat firat For lince Jut, lMlCtion aeetion of the preceding chapter atatel .tate. that ~. ..,.. pmpoeition of the Jut pnIpOIition 00DneCted auIlee$ sublet of a R contaioa containa all pointe between any two of ita pointe. 00D1lected . . ,. 8inoe " Ie is bet,... between the pointe I(G), I(a), I(b) of /«(a, I([a, b», b)), we therefore have ., E laa, 6D. This Thia proveI proves the corollary. ,. E/«CI,

cat."" b».

b».

16. ,6.

UQtJaMCU I'UNcmONI IIJIQUllIICIIII OJ' I'UIICftONI

IS

Sometimes the name "intermediate value theorem" is applied to the slightly Olore continuous rea.l-vallJ4l(i real-val~ function more general statement that if Ij is a continuoU8 on a connected metric space E, then any real number that lies bet.ween between two points of J(E) j(E). The proof is of course the same saIne as a.bove. above. It. It I(E) is itself in I(E). is worth remarking that the validity of the intermediate value theorem theorern for all continuous functions on a fixed metric nletric space E is equivalent continuoU8 real-valued functioll8 to E being connected; for if E is not connected we can write E == A VB, where A and B are disjoint nonempty open subsets of E, and the theorem theorenl fails for the continuous fUllction function on E which is 0 on A and 1 on B. The previous us to give many n1&ny new examples exaruples of conprevioU8 theoreol theorem enables U8 nected subsets of nletric continuous image irnage of an metric spaces. }4'or I<'or exanlple, example, any continuoU8 open or closed interval of R in another metric space (a "curve") is conba.lls in nected. We now apply this idea to show that all (open or closed) balls Era, == (ai, (aI, ... , a,,) and q = (b1i , ••• , bit) bn) E"', and Era E" itself, are connected: If p .. are points of E", define the line 8egnlent between p and q to be the set of points legment betuIeen

I

(01 (aa

at>t, ... + (b{ba - a.)t, ••• , a" + (b.. (b. 1 -

Q ..

a..)t) : t E [0,1)) (0, 1]) a.)t)

E'-. C E".

Since the component functions tit + (b. ~)t are continuous, the line segfunctioll8 '" {b, - ",)t p and q is a continuous image of the interval [0, 1], h~ce ment between 'P h~nce is p and the point (aa connected. The distance between the point 'P (al + (b al -- a.)t, o~t, .• ", a. + (b" (b,. - Ga)t) a.)t) is t times the distance between p and q, for any ... [0, 1], hence at most the distance between p and q. ThU8 tt E (0, Thus the entire line segment between the center of any ball and any point of the ball lies entirely within the ball. Any ball in A'" A"'" is therefore the union of all line segments segnlents varioU8 points, that is the union of connected between its center and its various sets that all contain the center of the ball. By the second proposition of the prec«ting chapter, any ball is connected, last section of the prec~ing connected. Since E" Era is the union of all line segments between the origin and its VariOU8 various points, the E· is connected. same reasoning shows that A'"

+

+

16. § 6. SEQUENCES OF FUNCTIONS.

Definition.

Let E, E' be metric luetric spaces ~paces and for n

= 1,2,3, 1, 2, 3, ... . ..

let

I.:: E --+ - E' be a function. If pEE, we say that the sequence /I, I.. fl, I., ft, fa, ••• conuergu at p if the sequence of pointsjl(p),/I(p),/a(P), points 11(P), 1.{p),I,(P), ... of E' C01Werg88 /fJ' converges. functiona la'/.,/" We say that the sequence of functioll8 11, fl, fa, ... corwergu converge, on E, or confl, I" ft, "",U, or ia convergent, corwergent, if the sequence converges at each pEE. If la, I., .... converges and I: f: E --+ E' ie is the function defined by /" . . converge. I(p) f(1') = lim/.{p) lim/.(p)

ver,_,

-.-.00

/I, II, I" fa, la, . .. corwerge. I, I is called the limit for all pEE, we say that 11, converges to f, li'mit lunction function of the sequence, and we write f =: = ....... lim/.· linl .-ex» I.·

1M If

OOlft"lNU01Jll FUNCTIONS IV. OONTINUOl1l1 n1NC'I'IONlI

For example, for each 0n -- I, 2, 3, . .. •. let I.: /,,: [0, 1) -+ R be given by I.(z) /,,(z) - z - I:/n. %/0. For each :e %E [0, 1) we have lim 1.(1:) /,,(%) - z. %. Here the limit

.-,,-

function /I is i8 the identity function 1(1:) /(%) - 1:. %. This i. i8 iIlUitrated illustrated in Fipre 21.

I

FJoUJlll21. The eequence l8Cluenae of functioai/,,(z) Ftol1Bll21. functioni/,,(z) -

11& Ie -

11&/,. II. z/,. OD on (0, 10, 11.

For a second example, let I,,: (0, 1) -+ R be given by 1,,(1:) /,,(%) .. 1:". %". Since I.: [0, a" .. lal < 1 (cf. end of 13 §3 of Chapter III). III), the sequence /1'/ It,!.,•• lim o· ... 0 if Jol

.,,--

/., . .. converges to the limit function I/ given by I.. I(z) /(%) _

{O1 ~fIf 0zs S-~ I:%1. < 1

Notice that each each-/" is continuous, is not. Thi. This it is -I. it continuoUi. but the limit function it illustrated in Figure 22. If 8. B, E' are metric spaces la, .... 8p&ceII and the \he sequence of functions II, II. I., I •• I.. .. from 8E into E' converges to I, then for any Ee > 0 and any ,p E EE B there it is B' converpl I. \hen a poeitive positive integer N such that 4(j(P),I.(p» tf(J(P)./.(,» < e whenever 0> n > Nj Ni thie it previOUI definition. In general the integer N is a alight Blight amplification of the previous depends on both •e and ,. p, and for a fixed Ee we mUit must take N larger and points p if we want the inequality tf(J(P)./.(P» d'(j(p),/.(P» < Ee larger for dift'erent points, to hold for all 0> n > N. If it happens that for any Ee > 0 we can find an integer N that works simultaneously aU points, points pEE shall simultaneoUily for all E 8 then, then. as we ahall Bee, the convergence of fl' /1. fl' / •• fa, /.. . .. to If is especially nice in the aenae see, sense that if each Vf la, . .. po8tlt'JlI8eB pf the functioll8 functions II,II, II. / •• I.. POBBeBBe8 a certain kind of property (for example, continuity), continuity). then 80 does the limit function I. This motivates the definition on the next page. pap.

I

D4r1i.uticm. E, E' B' be metric metric..,..., ~nition. Let 8, ~, lortl for,. - 1, 2, 3a~t ... ••• WI.: let/.: ..... 11 B' /: B ... TheD tile ....... ...... be a function, and let I: - B' be another anotber funetion. fuDe&ion. 'l.'heIa II, eaid to COftM'fC c:onHrfC """...." /1, I., It, I., /., ... is said """'. ..... 10 lif, /if, atvatven any • > 0, . . . . . a tl(J(P),I.(P» < ........... lor aU poeitive intepr integer N IUch that 1(J(p>./. N, Nt for III pEE.

.

I., I., ... cony_ con..... uniformly If the sequence II, /l./t./••... unifonnl1 to J I we ... ........ ~ III.Y, for emphaais, •.• converpl con. . . . wniforntly eay, emphaaia, that 11,1 /•• / .. •• 1., /•••.. uniform., to/_ .. II. If the I .. ... to a certIiD ...... 8 01 • GOIIftIII uaifonqiy restrictions of II, /1, I., /t./., certainaubeet of II COIl. . . . uniI-..iJ we.y that/.,/../..... that 1,,/., I.. ......... _ B. to lOme some function on 8, we., ...,.". ~ ...

0IIII.,..... .................

J'lOVllll23. J'lova 23. Unif_

00II

fl • net ftIWIIIt.

,.... fIl . . . . . . . fIl ...............

16

IV. OON'I1NUOU8 CON'IINU011ll J1JNCftONI PUNCftONI

Uniform Unifonn convergence of a sequence of functions clearly implies convergence. The fint (z - ~/n) z/n) .. == ~z firat of our examples above, according to which lim (~

.ft-.clO

on [0, d'(/(x),I,.(x» -== (0, 1), is an example of uniform unifonn convergence. For here d'(j(~),I.(~» Ix can be made made lellS less I~ - (x (~ -- x/,.,) ~/n) I=-Ixl/'" I-I~I/n ~ l/ft, 1/11., and this lut Jut quantity can than any given n > N, &8 given.e > 0 by taking 11. H, where N is an integer at least as larp 88 lie. larpas1/e. However, in our 8eCOnd Beeecond example, which is the convergence of the liequence of functions x, I), we do not have unifonn uniform converz, z·, zI, zI, z:I, ... ••• on (0, 11, show thiBis to quote the theorem, t.heoreln, to be proved shortly, that gence. Oneway One way to toahowthisis the limit of a uniformly eequence of continuous functions functiol18 il is conunifonnly oonverpnt convergent sequence tinUOUI. tranIlate unifonn uniform convergence in the present cue case to tinuoua. Or we can translate mean that for any tI> > 0 we have z· < •e for all ~x E (0, (0,1), have:e" 1), provided only that 11. i. il 8ufficiently lufficiently larp, large, and if it happens that •e < 1 thil that" this contradicta contradicts the continuity of the function :e" x· at the point 1. AI 80 alao also with 8e(luences Be
s

I

ProJHMltlon. ~, tDitI& tDiU& E' com",., complete, tmd and lell.: let I.: E --+ E', E', Propoe.tloft. LeI, Let E, E' be metric melric~, , ",,1, 2, 3, .... ...• Then 1M aequence 01 junctiom 11,1.,/1, II, I., I., .... .. it 1,2,3, ollunctiona ia tmilorml1l unilormlJl ~ il i/. lor Oftll GAJI • > 0, 1JNr, poaitiH inUger integer H N eucA wcA tAcat tIuJt if and only cmlll if, II&ere it a poaWe if n and CIftd mare m care tIacan N II&ea (j.(P) , I.(P» < •e/O' lor aU pEE. ifn Feaur than then d' d'(j.(p),/.(p» p E B.

I

i,.,.,. ".".,. i""',

/1, I., /1, II, converpB unifonnly uniformly to I, /, then for any If the eequence sequence I" I., ... converpa exila a poaitive eI> > 0 there existl positive integer N IUah luch that d'(j(p),/.(p» d'(j(p),'.(p» < ./2 ../2 whenever n > N, for all pEE. Hence if 11., H, for all pEE we have whenever" n, m > N,

d'(j.(p),I.(P» s ~ tr(j.(p),!(p» d'(j.(P),/(P» tr(j.(p),/.(p»

+ d'(j(p),I.(P» tr(j(p),/.(p» < ~ + ~ == •• f.

This proves the "only ir' ir' part. We now prove the "ir' TbiI Hif" part: For any pEE, II(P),I.(P),I.(P), •.. ... is a Cauchy sequence in bi'. /,(P)./,(P),fa(P), E'. Since E' is complete, cODlplete, this functions /1, ,., fl' /., . .. conthia sequence baa a limit. Thus the sequence of functions/a./I, verpa. Let I/ be tbe the limit fUllction. Given •E > 0, choose the integer N 80 verges. that we have tf(j.(P),I_(P» tf(j.(p),/.(p» < 1/2 e/2 whenever n. ft, m > N, for all pEE. 11. > H I,(P), Then for any fixed n N and fixed pEE the sequence of pointa points !1(P), 1s(p),I.(P), ·..• auch that all tenns terms after the NIIt NtA are within diatance 1t
,6.18.

IIIIQUUCIIIII DQUUCIUI CW or FUIICI'ION1l ruMCftON-

d'(j,.(P),f(P» have tl(j.(,,),/c,» I(J.(P),I(P» S e/2. t/2. Hence if "ft > N we ~ve I(J.(P),/(P» 'P E E, proving unifonn convergence. . pEE,

11 17

all < Ii~ for an

meb'ic.,.. _ let 11, It, I., Theorem. Let E, E' be metric apacea tJt&tl /1, /1, /., ... ••• be IIGuni/orml, uniformly conwrgmt continuouajunctiona into E'. 8'. Thtm Then limJ limJ",.. it iI ~ aequet'&CB aequence 0/ 01 COfttiftuoua /undiou from E i1&lo continUOUI. .-cD continuoua. ..-

.>....-

We must ,,. is continuous at each point po pu E E E.. muat show ahow that , I- lim I. ia continuoua 80 let Po E be fixed. Let e• 0 be given. liven. Fix a positive integer n such PI E B I(J(P),J..(P» < ./3 that d'(j(P),I.(P» t/3 for all pEE, which is poaible possible by the uniform continuous at 'Pe, Po, there is a number a > 0 such that convergence. Since I. I .. is continuoua if pEE d'(j.(P),I.(Po» < ./3. t/3. Hence if pEE P E B and d(p, Po) PI) < a then d'(J.(P),I.
tl(j(p),!(Po» d'(J(p),/(pi» S d'(j(p),!,.(p» I(J(p),/.(p» e

+ d'(j.{fJ),I.{1Jo» d'(j,,(po),!(Po» d'(J.(P),I.
e

<3+3"+3"-" Thua I/ is continuous at 'PoThus

ahowa IOmethiDl The above proof really .hoWi IOmetmllI more pneral than is stated, D&lDeIy bave a sequence of functions fUncUODII/l, .. from 8E into E' namely that if we have /1, It, /1, I., /., .... that. converges uniformly on lOme open ball of 8E or center 'PI that, Po and if each I. continuous at ·Pt Po then the limit function is allO is continuoUl also continuoua continuous at PeIf J and , are functions functioDII from a metric apace space 8E into a metric space E', it is natural to try' try. to find lOme measure of the extent to which J and ,g diller, that is to find some lOme 80rt lOR of IIdistance" diiler, "distance" between I/ and ,. g. Fqr any pE E B we may say 8&y that I and, differ at p by the distance between specific 'P their values at p, that is by d'(j(p), d'(J(P), g(P», g(p», but we would really like to meuure how much I/ and, aU points of 8, measure and 9 differ over all E, not just at p. There various ways of doing this, depending on the circumstances are variOUl circuillstances and purpoeee in mind, but the most simple-minded method poees lllethod turns out to be one of the most useful. uaeful. It is to take the distance between /J and 9g to be

'0-

max (d'(J(P), (d'(j(p), ,(P» g(P»

: p E 81 EJ

if this t.hia maximum happens to exist. In order to develop this idea we need digreu for a simple lemma. to digress

let,

Let E and cand E' be metric metric: qacea, fuMLemma. 1M apacea, and let I and , be continuoua contiftUOUl/uftCtiona from E into 8'. TAm the real-tH.dued tioRl from B B'. Tlum real-tHJlued fundion function on 8 E whote woole IICIlue &taltMJ at poi'" pEE it tl(j(p), I (J(p), g(P» it contiftuoua. point ia continuoua.

u..

,(P»

on, ""11

88

IV. CONTINUOUS CONTINtlOUS Ft'NCTlON8 ""NCTIONS

We must show that this function is continuous at any given point Po (f (p), (1'), g(p» - d' (f (Po) , g{flo) po E E, that is that IId' U (/ (Po), g 0 for all l'p in some open ball in E of center fJo.

IId'{f(p), d' .(J(p), g(P» (/ (Po), g(Po» g(po) ) 1 g(p» - d' d'(f(Po), I

s

Id'(f(p), g(1'» - d'(J(p), d'(f(p), g(po» 1I + Id'(/(p), Id'(f(p), g(po» - d'U
., the last step being a double appli~ation application of the fact that the difference between t,vo two sides of a triangle is at most the third side. Since I/ and 9g are continuous at Po, po, each of the quantities d'(g(p), g(po», g(po» , d/(!(p),!(po» d'(f(p),/(1'o» is less t.han E/2 for all l' in SOllle center po, proving that for some open ball in ill E of cellter p in that open ball all pin Id'(f(p), g(p» - d'e/cpo), d'(j(po), g(po» g(po» 1< Id'(/(p), 1 < E, 88 as

desired.

In the case of greatest interest, that where E' := = H, there was actually d'(f(p), g(p» == no need for a detour to prove the lemma, for here d'(/(p), I/(p) 1/(1') - g(p) I and the continuity of this function follows from that of I/ and g9 in two easy steps: the difference of two continuous real-valued funcH is continuous. tions is continuous, and the absolute value function on R Now consider the set s: fronl E into E'. tf of all continuous functions from 88Bume that E is compact. Then it is true that for any I" I, 9 E tJtf We assume

max {d'(j(p), {d'(/(p), g(p» : pEEl p E EJ exists, since any continuous real-valued function on a compact metric attains a maximum. Hence we may define, for any I, g E ff, tf, the disdis-.. space attain8 tance between I and g 9 to be

D(f, g) DU,II)

= max (d'(!(P),II(P»: (d'(f(1'), g(p» : pEEl.

=:

tf, together with this D, is a metric space. We proceed to show that ff, tf, it is clear that D(J, D(f, g) ~ 0 and that D(J, D(f, g) = For all I, 9g E tJ, == 0 if and only if 1= I = g. It is also clear that D(J, D(f, g) :: = D(g,/). D(g,j). It remains to prove ooly tf then the triangle inequality, which states that if I, g, h E £f D(J, D(f, h)

5: S D(J, D(f, g)

+ D{g, D(g, h).

To prove this, pick Po E E such that D(J, h) = d'(f(Po), d'(I(Po), h(po». Then

D(f, h) = d' d'(f(Po), hWo» S g(Po» + d' d'(g(po), h(Po») D(J, (I (Po) , A(po» ~ d'(f{flo), d' U(PO), ,(Po» (g(po), he,,) ::::; (d'(f(P), g(P» g(P» : l' E EI + max Id'(g(p), {d'(g(P), h(p» : l' EEl ~ max (d'U
, + I

Thus ff tf is indeed a metric apace. It is "abstract" in the Rille senee that ita upoints" are functions on another metric space. "points"

18.

UQnN~

or nNOTIon

•

points in the metric apace epace •tJ is a eequence functioDi A sequence of pointa l!IeqUeIlce of or functioDl

11'/1'/1, / E. 11,1,,1., ... from EE into E'. This sequence will converge to a point point./EtJ if and only if

.-

D(J, I.) - 0, lim D(j,

in other words if and only if for each • that for any integer nft > N we have

poaitive latepi' > 0 there is a poeitive intepr N IUCh euch

E Ht < max 1d'(J(P),I.(P» Id'(J(p),I.(P» : ppEEl <.,I, that i8

d'(J(p),I.(P» d'(J(p),I.
<•

ThU8 the sequence of points .. of. for all pEE. Thu8 pointa 1.,1., 11, I., I., .... 01 tJ CODverpi convera- to aequeoce of functions II, It. the point II E If if and only if the sequence funetiODl/l' /., I., ... . .• on B converges uniformly to the function I. /.,1., /., ..• Suppose that a sequence of points pointa /1'/.,I., ... of. of If is a Ceueh, Cauchy aequeaee sequenee in the metric apace •• tJ. Then for an, any • > 0 there Ie a poaitive poeiti" in..... N IUOb. IUcb that whenever ft, ta, 7ft m > N we ha~ ba~

in,,-

D(j.,I.) D(J.,/')

< .. <"

i8 that ia

max (d'(J.(P),I.(P» : ,p E Bt BI

<" <"

or

d'(J.(P),/.(P» d'(J.(P),I.{:JI»

<•

for all pEE. Aeeume 01 tbiI Aaaume that E' is complete. Then the propolition pIOpoIition of section ~that the eequeoee aeetion is applirable, applicable, and it tells te1la us us'tbat eequence of functloaa fUDdioal

•

•

,.

IY. COIfIIMVOvalVllCIIOII8

/., /., /., ••• 80me function I:B-B'. I: B -+ E'. The 11,1 .. 1., ... converaconverpa uniformly on OIl B to 8Om8 previoul theorem teI1I UI that I iI continuoua. continuOUI. ThUll E If S and .,..mou. teUe UI tim/. -I 8m/.

....-

__ of pointe of the metric apace If. ThUi in the I8DI8 ThUl the metric space 8pace If ff iI is complete. W. have proved the followiDc foIlowiDa .... t. . We NIUI'-

com,., .....

na-r.",. _ . B' E' .......... .. ..",. . . . .,, MeA ctIMJII*. n........ lAl Lee B . teiIA B COfftPtM" .." B' coapWe. T . ,., .. 0/ GIl /WtdiorM /rotA B E 10 0&. tliaIortt:e tlial4ftl:e ...... ".". all eot&t,,""*, ~/wtt:IioMfroa to B', MeA teiIA IAe ., --Iwv:tioftll cantl , ,.,. Co be _'-~/_,"'to"

n. "'.,

,(P» : u .• ...". . . . . 0/,.,. U . ." . .... .,... A . ........ 0/"." 0/ IAU tMric.,...,..,.,.,.. fINtric . . . COftI1trIWI if .." .." if Jww4ioft1 ... B. if.."..." if"il u. Ufti/..." ~ co..,..., CIG....,.., ........ 0/ fwdiou max (I(/(P), (tl(J(P), f(p» : "E p E BI. BI,

Oft

If B' - R .e metrlo .... GI of all continuoua oontiDUOUl ...... real-valued U II' we haft the IDtRio ftlued IUDotioDi clIatance between two IUCh such fuDot.IoDII on OR • a compact comPM' metric . . . . B, the diMaDce fUDCtloDlI and and ,, beiDa beiDa max II/C,) ( I/(P) - ,(,)1: ,(P) I : ,p BJ. HI·

'''De/

e

TbiI ill important eDOUIh to be cIeDoted clenoted by a atandard ltandard I)'IIlbol symbol ThiI metric apace iI impoa1aDt eIlOUIh C(B)• C(B)•

........... n-..

. .OIlLDlI 1~

DiIcuII the aoatiauit1 fuDctioD I: ..... 00Iltiaui\J of f1l the fuDetioD • - • if JO if s
1l.

(b) 1(8) (b) /(.) --

••! •• ! 1f.,.0 if . .·O { •s { o 1.-0

(....... propIItIeI f1l ( - . e the ..... propenieI of the IiDe line luactioo fUDCtion .... are kDowa) Imown)

(0)

.--. 1 1o IC/I) - ~! {

IC/I)I~)-

(cI) /(11) (d)

ifs .. O if O 1 Is-O ifs-O lis Ie luohatloa" if. DOt ratioDal

_,1

,

Is if • -_

r, ....... , .... ___ r..... aDd ,, .... are ia la.... , with DO COIIUDOIl COIIUDOD divllon cliviion ott. tbaD tbua .1, *1, .... other ud ,> f > O.

PROBI&MIJ PBOBLBM8

91

2. Let E, H' I: E -+ H' E' be metric spaces, 1: E' a continuous function. Show that if S 8 E' then J-1(8) subset of B. E. Derive from this is a closed subset of H' )1(8) is a closed 8ubset the results that if 1 is a continuous real-valued function on E then the sets :/(P) ~ OJ, (p E B: E: J(P) I(P) S ~ OJ, 01, (p E E :I(P) 01, (p tp eE E E::J(P) I(P) - 0) 01 are closed.

E' be metric spaces, 1: E' a function, and suppose that 8 8,,8t 3. Let E, H' J: E -+ B' 1,8. are subaets 01 of E auch that E - 8. U V St. S. Show that if the restrictions of 1 closed sublets B such / to 8 1 and to 8. are continuous then I/ is continuous. f. ut Let U, V be (open or closed) intervals in R, and let 1: 4:. J: U -+ V be a& function X,1I 11 then /(x) I(x) < J{1J» I(Y» and which is strictly increasing (i.e., if x, 11 E E U and x < " onto. Prove that 1 J and,-· and 1"1 are continuous.

5. Let H, E, E' be metric spaces, 1: E E. Define the f: E -+ E' a function, and let 'P pEE. oacillation 01 011J at 'Pp to be 0ICillati0n g.l.b. (a E R : there exists an open ball in E of center p such that allY z, X, y in ill this thill ball ban we have d'(/(x),!
.... --

x ~ at, aI, and where 1+ is the restriction of 1 I to U "("\ (x (z E R : % limJ(x) limI(x) .. - limJlimI_ (x),

........~

....

where 11_ is the restriction I to U (\ J,I if these limits exist. rel!triction of 1 ("\ I(xx E R : xz ~ a 1 Prove that lim 1 / exists if and only if lim 1 / and lim 1 / exist and are equal. -• ..... + ~+

~ ~

.... .~-

aJ, for some positive real number a, and let I be a 8. Let U - IIxz E R : zx > ai, real-valued function on U. Define limI(x) lim g(lI) g(y),, lim I (x) === linl

.-+-

...+-

.....

....

g: (0, l/a) where ,: Ilea) -+ R is given by g(1I) g(1/) ... =- I(l/y), /(l/y), if this latter limit exists. lim/ex) exists if and only if, given any tf > 0, there exists atnuma,numProve that lim/(x)

...... -+<0

II(x) --/(1/) I(y) 1I < f. > N then I/(z) v"i is continuous on IIxz E R : z 2: ~0 OJ. I.

ber N such that if s, :t,1I X,1I her 11 E R and x, fI 9. (a)
V"i-l

!!'!!.

(c) 2.zI ~ 1 (cf. Prob. 8). (0) Evaluate .1i~ 2zI

92

FUNCTIONS IV. CONTINUOUS FUNCrION8

10. Di!l('uM ...... R if DiACUflS t.he continuity of the function !: /: E' --+
=

s::

(b) !(x, I(x, 1/) f/) -

. (e)

(c)

_ l _ if (x, 1/) f/) '" ~ (0, 0) _1_ { xx't +0 1/' tit { (0, 0) if (x, 1/) tI) - (0,0)

~ if(%,y)~(O,O) -EL if (x, 1/) '" (0, 0) { x' +0 fI' ~ +011'

{

if (x, 1/) II) - (0,0)

{%71' {x1J' I(x, y) =- st !(x,1/)'" ~ +,+0 11'

if (x, 1/) ,. 0) '" (0, (0,0)

o

(0,0). if (x, 1/) f/) - (0, 0). 11. ,3 (on the continuity of SUIDI, sums, prodJl. Give a proof of the first proposition of 13 continuous functions) that ia i8 hued balled directly on the definition of ucts, etc. of continuou8

continuity. functioDs 12. Prove the analog of the first proposition of 13 ,3 for complex-valued functions (cf. Probs. 20, 21 of Chap. III). on a metric space (ef.

following alternate proof that a continuou8 continuous 13. Write down the details of the fonowing E is bounded and attains a real-valued function function!I on a compact metric space H H maximum: If! If I is not bounded, then for" - 1, 2, 3, ... there is a point p. E B (P..)II > fI, ", and a contradiction ariaes such that If (P.) arises from the existence of a convergent subsequence of Pt, . . •• Thus! Thus I is bounded and we can find a PI, Pt, Pa, PI, .... sequence of points q" lim/(q,,) J.u.b.I/(P) :p E E} .• ql, qt, q" ql, q., ... of E H such that lim!(q ..) .. -l.u.b.(f(p) EEl

..... .-

by! subsequence A maximum will be attained by I at the limit of a convergent subeequence q., qt, q" qt, qa, .. •••• ·• of q., 14. (a) (a> Prove that if 8 is a nonempty compact subset of a metric space H E and Po d(po, p) : 'P 'Po E E then min IId(p., p E 8) 81 exists ("distance from PI 'Po to 8"). (b)a B- and 'Po Po E EB- then (b)o Prove that if 8 is a nonempty closed subeet subset of Emin (d(p., 8) exilta. Id(p., p) : p. p E 81 exists. space H, max I(d(P, d(p, q) : P, 16. Prove that for any nonempty compact metric apace p, q f E HI g,,). (Hw: pointe exists ("diameter of E"). (H"": Start with a sequence eequence of paira of polntl I (P .., qf.) ..) J.. \ ..-I•••••••. -1 ........ of B g such (Pa, woh that 11m d(p", d(p., f.) q.) - 1.u.b. P, q E BI H\ lim tu.b. IId(p, d(", q) : J',

....

,,-

and pass to convergent subsequences.) 16. let E, E' be metric 1p&Ce8, I: H B -+ B Let H, spaces,!: - E' a continuous function. Prove that if H is compact and I is one-one onto then !-I: I-I: E' -+ and! - E is eontinuous. continuous. (Hint: IJ sends closed sets onto closed sets, therefore open sets onto open seta.) sets.)

17. Is the function zI ~ uniformly continuous on 81 R? The function

v'TiT? VTiT? Why?

E, the identity function on B E is uniformly 18. Prove that for any metric space H, continuous. 19. Provt' Prove that for any metric space E and any Po E E, the real-valued function d{7Jo, 'P) p) ia is. uniformly continuous. sending any pinto d(po,

20. 3).

State precisely and prove: A uniformly continuous function of a uniformly contin'ous. continuous function is uniformly eontin'ous.

9S

PIIOBLIlII8 PROBIAB...

2I. Let S be a subset of the met.ric space E with the property that each point of 21. eS is a cluster point of S (one
22. Let V, V' be normed vector spaces (d. Prob. 22, Chap. III) and I: V -..... V' a linear transformation. Prove the following statements. (a) If 1 is continuous at one ~int point it is continuous continuoul everywhm'e, everywhere, and in fact is
0,

Z.,

(Sa lit, Zt _ + .,•....) (Zl + JlI, III, %t Zt + 1ft, II., ••• ) Zt, :ra, ... ) - (cst,"" (c.zl, CZt, CSa, c.za, ••. ••• )) Zt, ZI, ZI, ... ... )11 - max max {I Isal. Izal, Iz.l, IZtI, ... 1,I, II (ZI, %I, IZII, I-I,

(ZI, %1, %a, • .)) Z" ...•

lit, 1/., 1/1, ... • • .)) -+ (y., (III, II"

C(ZI, Zi, Z" • • .) C(Zl,

and the map sending (Zh (ZI, ZI, %t, ZI, ••• ) into (sa, (ZI, 1st. 2%t, 3za, 3%t, •••) ••• ) Ie Is •a 0De-0De oae-oDe normed ~tor !pace apace onto it.lf itllelf that that. II Is DOt not. linear transformation of this Donned ,. continuous. ~ space O~ over II 23. Use Problem 22 to prove that if V is a finite dimensional vector apace R and 11111, II lit, 1111. II Itt are two norm functions 01\ V (i.e., real-valued functioDl functions lOch IUch IIlh) and (V, 1111' Vector spacea). apa.cea), thea then there exiat exist poIipcMithat (V, 1111.) II lit)) are normed vector that. m S IIzIlJllzU. DOnsero •a E V. tive real numbers m, M such that 1I~IIJllzU, S M for all DOIlS8fO normed vector apace Deduce that any finite dimensional Donned space is complete (u a metric space).

e v.

24. Give another proof of the intermediate value theorem by compIetiDI completing the folcontinuous real-valued function OD on tile the cloled cloeed iDler· interlowing argument: If 1 is a& continuoul (II, b) bl in R and 1(0) 1(11) < 'Y < /(6), I(b), then val (0,

Is E (a, II) : 1(%) -,1) - 'Y. /(1.u.b. {z l(l·u.b. (II, hI: fez) S S 'YI) uaiDi uDiform uniform ooDtlDult)'. oontiDuit),. 25. Give a proof of the intermediate value theorem Ulilll URing the notation of this theorem. theorem, uniform aontlDult, aont.inuity lmpU. implitl that. that, (Hint: URinl given any e > 0, if we divide [CI, alUftloiently Iarp Dumber . [a, 6) into .. auftloiently larp number of ..biD. bbatervals , .e Viols of equal length then for at leut least ODe one 01 the division poiDta points, we IhaIl have 1/<1') 'YII < e. Choose a sequence of ,'. ,'s correapondinl corresponding to a.. sequence ,_ I! (1') -- -y aequeoce 01 of i. approaching approachiDg zero, then a suitable subsequence.) II, b hER, b, and let 1 I be ..a continuous real-valued rea.l-valued function OIl on (a, lo, 6). hI. 26. Let 0, E H, II 0 < h, onlH)ne then I«(a, 6) is (/(8),/<,» (J(a),I(i)) or (J(b),I(a)}, Prove that if 1 f il is one-one /«(0, 6» (/(6)./(0»), wbiebever expreaaion makes sense. expressioD

27. Show that if I: R ..... -+ R is a polynomial function of odd degree, tben/(R) thenl(R) ··R. 28. Show that any open or closed interval in Era E" is connected.

metrie space E is said to be or""... 1IT1'tIr188 CORMdMI emmect«f if, given any any" 9 E If, there 29. A metric p, fell, is a continuous function I: (0, 1) ..... /(0) - ,./(1) - E such that 1(0) ,,1(1) - f. Show that (a) an arcwise connected metric space is connected subset of EEta is arcwiBe arcwiae eoDDeCted. connected. (b) any connected open sublet

M

IY. OONDNUOua CON'l'INUOua rvNCl"I'IOH8 rultCTIOHS

30. Prove that a continuoua continuous realZ:"-t' ~\.aiued cloaed interval in .. gt eancan80. ued function on a cIoaed DOt be be ODe-ODe. ODe-ODe. ,1 DOt .' 11. t· 31. (A ~ apace-filling curve) t' ' (a) (a> Show tba, tha~ the sublet IUbeet 8 01 (0, 1) I) CODSiBtinl consiatinl of all numbers having decimal expanaiona of tbe the form expaDIioDa 0

.tJ.btca~.J) •• •••• .(JI6tcl~1J Ga, 6., 6", c. beiDa bein& one of the inteprs 0, 1, ., dOled. (each a., ••..,, 9) is cloeecl. (b) Show that the real-valued functions .,., f'a, cpa, IPa, fit IPa on 8 which send

•• a~ ••• '.I~'" numbers with decimal expanaiODl expanaions into the real Dumben

.6a6s6a. baba ... CaClC. ..••• . .•, •.CaCsC.· respectively are continuous. (Note that each number in 8 has a& unique that fPI, f'I, 'PI, are well-defined.) decimal expaDlion, 10 \bat \fI,"'t.pat (0) Show that there are unique continuous OODtiDuOUB real-valued /., /.,!. rea.l-valued functions la,I •.!. on (0, 1) whOle whose restrictions to Bare"" 8 are f'a, -PI, IPa, '" IPa reepectively, respectively, which equal 0 at 1, and which 6, Chap. whieh are linear on each open interval in e8 (cf. Probe Prob. 5, III). (d) Prove that the function I: 1) ..... B' defined by I(z) - (f. (z),/I(s), /: (0, (0,1) - BI (fa(z),I.(z), 1.(%» (0,11 ill a continuou8 continUOUIl map of (0, (0,11 /.(z» for all z E (0, 11 is I) onto the unit cube • tJafIttJI •••• .CJaGtClt • .,,

°

I (Za, %a, ZI, z.) E E' : Zi, ZI, :&a %a E (0, lO, 1) 1),. Za, %a, I. functions 32. Show that the sequence aequence of funet.ioDa

V"i, *.Jrz-+-V-;::z=+="'::;;z, ... ..• Vi, V vi zz + vii, Vi, ..;rz-+-"-'=z=+=Vi==z, 08 OJ is converpnt and find the limit function. 011 Is Iz E It: R : z ~ 01 functions S, z, Sl, z', zI, ..• converges uniformly 33. (a>
z

'

.~

u u

I.(z) - 1 + ftS'? /.(z) - 1 + ftia? /.(z) - i"t n,zt? If I.(z) ,.it? If J.(z)

u

+ ,.Iza? "lzlT

36. Show that if the function I: R .... ~ is uniformly continuou8, then the sequence --. R

36.

n,

1),/(z O,/(z n,1( + ~), ... is uniformly convergent. Does the lequence j, i,i, Bequence of functions z, ~, i, i, ... converge uniformly on 81 R?

functiona/(z of functioDa I(~ + 1),/(~ +

%

37• /., ... and 'I, • •. be uniformly convergent 37 • Let /t, /a, I., /1, I., la, ft, It, fa, It, ... convercent sequences of realyalued valued functions on a metric space B. E. Show that the sequence /. la + '1, la, /. + ,.,/. .• is wliformly /a '1, It 'I, '1, ,. It, /. + la, It, •... uniformly CODVerpnt. converpnt. How about la II, II I. 'a, It, ... ,T 3L 38. Prove t.bat that the limi~ limit of •a uniformly convergent convercent sequence aequence of bounded functions (from one metric apace into another) is bounded.

D. aD example of •a OOIlverpnt aequence of continuous real-valued at. Olve Give an converpnt sequence I\IDOUoDi on whole limit function faile fUDGtiona 011 lO, (0, 1) whoee faila to be continuOUl continuous at an infinite infbIite aumber awnber 01 pointe.

no.LIIIIS

9S

40. 1M a, • E .,.
'1.

-....

/1, It, /., ... . .. COOverpll CODVeraea uniformly. ,p E B B then the sequence aequence 11,/.,/., G. baD in C([O. radius 1 is not compact. a. Show that the c10eed cIoeed ball CnO, 1]) of center 0 and mdiUl (Hint: aequenC8 of Z, st, zI•..••) (H"": CoDlid. CoMicl. the aequeoce or functioDB fUllCtions #e, zI, zI, •.••) "S. Po E function F on a. HIf IIB iais •" compact metric apace and aod ,. e B we pt "& real-valued fUllCtion C(1I) F (f) -I (PO) for aliI all I E C(8). C(B>. Prove that F is uniformly conC(8) by aetti. I18ttiDa F(f) -I{JI.) t.inuOUI. tiaUOUL ... GeDeraliae Problem 43 a aa u folio .. : H B aod 4oi. Geben1iR foUo..: If II and B' are compact metric ap&ceI spaces and fI: E ill •" continuous C(8') into C(8) C(B) by I8Ddiq sending each ,,: B .... - B' .. COOtiauOUl function, map C(B') Ie continuous.• IE 0(8') C(B,) into iat.o 1 I 00 " E C(8). Prove that this map is uniformly continuous .a. C(E) is a complete comp~ete normed Donned vector .u. !At IAt B be a" oompact compact metric apace. Show that C(8) apace (of. Ptob. 22, Chap. III) if we add it. ita element. elements in the usual way, way. mulPtob.22, tiply real DUlDbenJ·in uaual wal, way, aDd and take 11111 tip1, them by ..... aumbens'in the UIU&l HJU - max 11/(,) IIICP) II:: ,, eE BJ 44 is "a cootinuous continuous &'1 for aD aliI/ e C(8). Show that the map of Problem .. linear liaear traDlformatioD. tnuaIIormatioo. 4.&. Prove the &D&1oI ualog of the Jut tbeonm 48. theorem. of the ohapW chapter wbeD whell B is DOt compact but, restriction to bounded ooatiDUOUB functioDl, the cliataDce distance between but with a ..wict.ioa COIltiauOUl functions, IUch fuDctioD8l functioDl/ and ,f beiDc heiDI taken uas . two BUell

(cr.

1.u.b.(cI'(j'CP),fCP» : p 1.u.b.ld/(J(P),,(P»

eBI.

aame thing tbiD& for bounded functiona functions from &' Do the lime B to B' that are not DeCeIIIaI'ily neceaaarily continuoua. What is the relation bet.ween continuous. between the two metric ap&cell1O spaces 80 obtained?

CHAPTER V

Differentiation

The subject .. IUbject, Of of thia tbia chapter is one-YUiable one-variable . M . The eMeDtial eIIeIltial items, ..,. tMlr ential ealeub.. caleulua. ne item8, and eva Uaeir are familiar from elemeBtarJ elemet.t• ., caIcuI.. caIeu1ua. development, arelanuliar Thil preaiIion This pound IfOUnd ean can ~ covered oovered with epeed tpeed and pnaIeion since all the difiieuJt . . ...... .. sinee difiicult work hM been clone doae .in. tile

cedina chapter. cedins

.

•

Y. DIrn:••N'IU'ftON

11. II. THE DEFINITION OF DERIVATIVE.

DejinJtion. subset U of R. lhdiftltlorl. Let I be a real-valued function on an open aubset Letz.E say tbat thatJiadi6~.Stif Let %I E u. U. We We.y I is dil~ at z, if I(s) -/(-) -/(~ lim [(s) ..... ....

~-St S-%I

exietl. (Zea), is called the tIeritHJliH derivative 011 exista. If it exists, exiBta, this limit, often denoted I' I'(~, atZtat %to

We remark to begin with that t;1 t~notion notion of limit used here is exactly u..t of the preceding chapter, since' that since'~ a cluster point of the metric apace U and we are conaidering a function!trom el%eJ of 1%11 IZlt function:ll'Om the complement el%ll in U into the metric apace R (namcay -.ociatea to (~y the function which UIOCiatea el%ll the element (j(z) (f(z) - /(zO»/(% /(z~)/(z - z.) each z21 E elz.1 %0) of R). AIJ As alwaYB, alway., if the lintit (Zo), if it exiata, exists, is neceasarily nece&8&riIy unique. limit exists exista it is unique. Thus I' I'(~, A clearly equivalent definition 'of given by of I'(~ is liven

-10m.

J'(~ /'(~) _ lim Jl%t'+ .a~.+ A) - /(~ J(~) • A...e ~. II

"

~,

Here II" is ~n~entood o. tn"entood to vary in 80tH Open ball in R of center O. Theeiuation The"tion

'.

I(z) .... /(~ .. _I'(~ lim /(s) ~ /(zi) /'(%t)

....... . z-:-. z--.

equiv4t to the existence, for is equiva.t

ea4 Ie > 0, of a number 3a> 0 luch that

~~ II/(:I:~ =:-/(~ /(Z)

Z

"Zo,

'J'(~I -z, • ''':'I'(:r:.) ,

IS~ • I

'

121 ~zj 3. The last whenever z21 E U, z ,. So, and Is ~•. <
1/(1:) lI(z)

·.t'.t

-!(Zt) -I'(~)* - %I) IS -/(~ -1'(Se* IS

.Is %II,, 'Iz - Sel

alao holda holds if z - Zt. %to Note No. if •I is amall enouah which also ia taken 101a1l enoulh and 121 - Zli %II < a • then it is automatically., automatically" that Is tbat 21z E U, Bince since U is open. Thus we can .. y. somewhat Illore more briefly, that tbat the equation say, lim 1(:1:) -/(Zt) -1'(%1) _ /'(Zo) /(21) -/(z.)

....

s-z, s-z.

ill equivalent to the existence, for each Ie > 0, of a number is nUlllber 3a > 0 Buch 8uch tbat that

I/(s) I/(~)

Iz whenever J~

elz - %II :r:el

-/(~ -I'(~(z S 1121 -/(~ -1'(St)(~ - ~ Zo) IIS

Sel < •. a:ol

11.

DJUlNI'I'IOIf 01' DUlVA'nVB DUlVATlV& DUlIII'IIOIi

"99

R- a B ia Recall that a& function .,: ,,: a is called linear if there exist numbers .,(z) ,. (p(z) ,. c,l: c, A: E a R such that .,(s) - aes + A: for all sz E a. R. We then have .,(z) .,(z.) c(z - z.) • real-vaIued ,,(Zt) + c($ Zo) for any :re .%0 E R a.•'Dle real-valued function on R sending any z into 1(Zt) Zt) it followa that I it u dil"'~ diJ1erentiable I(z.) +1'(Zt)(z I'(z.)(z - z.) is linear. It foll01fl at if and 0IIlJ ~ ,..,. /tmctior& GC :re Zo il 0IIlJI ill if I ca CM IN be t:loMlI doad" G~mot«l MJr :re .%0 br brI G liMClr lifteGf' Junctiora in the I8DIl8 __ that there then exists a& linear function tha~ ditlera differs from IJ by & a am..l fraction of Iz -:reI is sufficiently near:rei very small -.%01 if z it near Zo; worded preciaely, precisely, this condition is that there exist a& linear function "I{J 8uch such that for any e > 0 there exists a 3 > 0 such that

+"

I/(z) whenever Iz -:reI -.%01

e

.,(z) 1 I ~ elz tlz --.%01 z,1 ,,(z)

< 3. a. , -/(z) - f(%)

VlQu•• 26. 25. Graph of 01 •a funetiOD fUDetion that is differentiable at z.. Ftou•• Zo. Near z. the graph ia i~ very cIoIIe eertaiD .traicht straight line c10lIe to •a eert&in liDe (the "tanpnt line at z% .. = z."). Zo"), in the IIeIlBe eenee indicated.

ProPfnltion. be an BUbBet 01 a,l: lJ - R. III is diJfere,&tiable dijfere,.tiable Pro"""ora. Let U IN CUI open aubtet R, I: U III it at E U then continuous tit at z .. GC :re .%0 E t1um I it is cominumu Zo-

u

Pick any ft eo > 0 and then a 8uitable suitable number Ie 30 > 0 8uch such that I S tvl I/(z) -/(z.) - f(zo) - f'(z.)(z f'(Zo)(z --z,) --.%0) IS eol z - z.1 Zol whenever Iz -.%oJ - z,1 < 30. Then if Iz --.%01 %01 <" < Ie we have If(z) --/(z.) 1(.%0) 1 I/(z) - 1(Zt) - f'(Zt)(z --.%0) zo) I I/(z) IS ~ I/(z) I(z.) -I'(z.)(z z.) 1+ II'(zo)(z I/'(Zo)(s - %0) S (eo + (.%0) f) I) Is Jz -- :reI· .%01· ~ (tv + 11' If'(z.) e/(ft + f)1, If, for any an)' •e > 0, we choose 3a - min (a., ./(.. + 11'(z.) II'(Zo) I) J. we have I/(z) --/(z.) f(zG) 1< I < •e whenever Iz - .z,1 %01 < a, 3, pro~nl proying the propoeitioll. proposition.

a..

180

v. ....

DlITSUlftUftON DlJ'l'JlUN'lU'II01l

Dfdinidon. real-valued lhrIbddoll. Let /J be &a .... -vaIued function on an open IUbiet aubeet U of R. So E U then I/ is called diI tli6~ on U (or Jun just If /'(J:o) /'(s.) exiat8 exiata for all &11 a:. ....diable "" di,jfmmtitJblB). /', olten often denoted tl//tk, tlJ/ds, ill derivative dillIrfIftliablt). The function function/', is called the tleriwtiH

all. 011· The notation d//tk dJ/tb; (or d/(z)/tb) has many obvious defects, but at tl/(s)/tk) baa least we usually know what is i8 meant. necessarily continuoue, continuous, but a oontinuoue continuous A differentiable function lunction is nectlllll&l'ily function i8 is not necessarily neceaaarily differentiable. For example the abeolute value function is continuous on all 01 of R but it is Dot since if not differentiable at 0, ainee zP'O s"O

Isl-IOI s - 0

-

W. {{ .l!l sZ --

1 if s > 0 o. -1 if s < O.

No limit can exi8t op8n b&1I ball in R of center eenter 0 exist aa as s2: approaches approachea 0 since any Op8n numbera INAter contains both numben greater than aero zero and numbera numbers _I. . than MI'O. sero.

II. t 2. R1JLES RULES OF DIFFERENTIATION.

In very simple C88e8 cases it is easy to differentiate (tbat is, compute denvaderivativea)' directly from the definition. For example, if J tives)' / is a conatant constant function, that is if fez) - c for all 2: il J(z) s E R, where c is some lOme fixed real number, then a:. E R we have for any Zo J'(a:.) - lim J(s) J(sO) == _ lim ~ - lim 0 - O f'(%o) :a /(2:) - !(z.) O.•

..... ....

z-z. :r:-a:.

..... :r:-:r:e z-z. ....

.....

g(2:) -:= :r: Z for &11 all :r: z E R, then for any If IIg is the identity function, that is if II(S) zaER we have xoER 1I'(a:.) -lim lI(s) -1I(a:.) _ lim :r: g'(Xo) ::II lim ,(z) - '(%0) == z -:r:e - Zt _ _ lim 1 _ - 1.

- ..

.....

Z-Zo :r:-So

z-z. ...... :r:-:r:e

.....

-..... ..

reaulta are usually uaually written These results

.-0, .-1. de

tk th

For more complicated functiona, differentiation by direct 1'8COUI'I8 recourse to the definition is impractical, 10 80 special rulea rules are developed. The followina followinl proposition makes the differentiation of rational functiona functioDl almoet almost mechanlor differentiating exponential, loprithmic logarithmic and tripotrigonoical. The fonnulas for functiona will have to wait until the next two chaptel'8, metric functions chapten, where theBe these functions are given adequate definitions.

,2. aVla am.. 01' ~ftO. DIJ'nIIIIII1U'1'IOM 12. 101

PropoRtion.. 1M cmtl , be "~/tmt:Aou ~ ItIfIdiau on em -_ " , . ...,., U 0/8. 0/" PropoaJeJon. U/I tmtl, II J tJRtI / + I, I,. 111 cmtl , til. care diff".entiable di6ereMabit ,., eM 1M poiftC Zt Z. E U, lAM •eo ... 1 " 1- I, If, Ofttl, ~ 0, 1/,. J/,. TAN tlerivativel derivativel ,., at %0 Zt ar. . . btl. cmtl, if il ,(So) '(%0) .,. are , giNn '" 1M ,....,., I~

't

(f+ (J + ')'(%0) ,)'(St) (f ,)'(Zo) (J - ,)' (Ze) (f,)'(~ (J,)'(z.)

-'/'(%0) (~ -' f'(s.) + g'(Zo) -1'(%0) - " (~ -I' (z.) (zI) ,(z.)/'(z.) --/(%o)(z.) J(:I:t)"(~ + ,(»,)/'(->

L)'(St) (%0) _ (1-)' ,

+

,(:t:e)!'(%t) -/(~(-> '(%0)/'(%0) -/(~(z.) •. (,(~)t (,(-»1

com~OD. TIae The Umi' Omit The proof, to be given shortly, is by direct computatiOll. formulas formulu of the corollary on page 76 are ueed wsed npeated1J. repeatedl)'. The ooatiaUi'J eont.iDai\y of 1 I and , at Pt of the etatementl fit is al80 also used, in the form 01 .tatementl lim I(z) --/(%0), ,(z.). limJ(z) /(%1), lim Jim ,(z) - ,(a,).

-.....

-

....

,
ease of the function 1/" the BMUmption 888Umption that ,(z.) ,. 0 inIIuN In the eue iDIUNI 01 center _ z. (by the eontinui. eontinui\y of , at ,(z) ,. " 0 for all &11 z in lOme open ball of z. and the reIU1t reeult on pap 76), 75),10 I'eItrid U to •a 2:t 10 that it is permieeible permilaible to I'eItri.d 8Ill&Iler set containing baniBhiDl all lDl&Iler open let ~ntainilll %0 Zt on which , is nowhere 1Iel'O, RIO, banieNDI concerns about poBBible aero denominators. denormnatora. Now that we have liven given all the pSble lero re&IOns for the validity I'eaaons v&iidity of the fonnal formal proof., proo", here are the formal proofs plOOfa themeelvel: themselvel : (f -/(~ - ,
The proof for (J (f - ')'(%0) is the same; jUlt jUit replace each , by -,. (/,)'(z.) -_ lim I(z),(z) - !(%tl,(z.) I(z.),(!!) (J,)'(z.)

..... -..

z-z. Z-%o

,<»,) + ,(z,>[(.) ,(z.) I(z) --/(~) I(N) Z-Zt

_ lim (f(z) (/(Z) ,(z) - ,(!!) ..... z-z. .... ~-z.

. - ..

_ Hm/(z) lim I(z) •-lim lim ,(z) --,(~ ,(%0)

.... .....

...•• --.

-/(%0)(%0) -/(z.),'(Zo)

z-z. Z-Zt

I(!) -/(N + ,(z.) ,(St) • Hm /(.) -/(~

-j"

...... --. z-,Zt

+ ,(z.)/'(z.). ,(z,)f'(t:e).

To find (//')'(%8), fint find (1/,)'(_): (//')'(%0), it i. is a little easier to 8rat (l/,)'(z.): (1/')'(%0) - lim (l/r(z» (1/,(z» - (l/r(a:e» (1/f(z.» (1/,)'(_) .... z -Zt ..... z-Zt

--

8m lim ,(z) - ,(Se) ,(z.)

..... == _.....

sz-z. -So

.....

,(~ lim ,(z) f/(s)

f(~ -_ .... lim ~ _. )) - .. (s (z-

__ _ _

(a-.) . (s.) ('(z.»' (,<%f»'

1.

Y. DJI'nUNTIA'IIOIl

Therefore our final step is U/,Y(z.) (J. (l/,»'(Zo) -/(Zo) • (l/,)'(Zo) U/,)'(Zo) - (j.

__ 1(%o)g'(Zo) /(%I)g'(%I)

I'(xo) = + /'(Xo) ...

(g(Zo»1 ('(%1»' CoroUary II I/ ia ill Corollory 1. 1/ cER, tAm

G (J

'(%0) ,(Zo)

+ (l/g(Zo»

'1'(Zo) ·/'(Zo)

g(Zo)/'(Zo) -/(Zo),'(%o) -/(xo)"(%I) (g(Zo»1 (g(Xo» ,

retJH1Glued junction on •em open aub.tet aubaBe nol-txJlU«l function

01 0/ R

mad and

This means, of coune, that for any Zo E R at which I/ is differentiable, (cf)'(Zo) (cf)'(~) exista and equate equals cf'(Zo). cf'(~). This followa from the formula for differentiating a product, together with the known result that the-derivative the-oderivative of a coDltant function is 1eI'O. aero. CODltant

Corollary J. CoroUary

For emy integer ft, n, d,r
It is ia undeJ'Btood that if ft n ~ 0 then the function Z" x" is defined only on nonseJ'O real numbers. n -.. 0 or 1. If ft n is a positive the DODJI8I'O numbeJ'B. The result is known if ft intepr greater than one we repeatedly apply the formula for differentiating a product, as followa: dzI tI dst d . (x •• z) x) • . - .dx (z dst d dzI tI •'iii" -- 'iii"(x .(21 • sI) zI)

dz dx

dz dx

dx ... -- ~ ~ + x~ - Zx •• 1 + zx ...= 2z dst dx dzI ';; dz - Z x dx x •b xl -... 3sI ax' dz -t 21 2z + sS iJst tI dzI, dxC d dst '. a dx ax' + +x'4:r;I •dZ - 'iii"(x' a (21 • sI) zI) - ~4i.7 ~ zx'• 3sI sS - 4s' dst d dxC' dzI tI . dz" adx 'iii" xt) - ~ + ~ - zx • 4s' 4:r;I + :r:t xt .. ... 5xt. di'" - 'iii"(x di'"(s .'• sI) 5:r:t.

+7 -

+,7 7 -

pl'OCe88 can be continued indefinitely. Each computation works Clearly thia this ptocela worka out as above, giving at each stage the formula d,r 0, and complete the proof with the computation

dl til

dxtlZ'"

-mx--

s-. a - 1 • di' -mr-I ~ 1) X- • di" dZ 1 1 (s-)' == sS. - -mz-o.".. - C&\z;i' di\~ (x-)' xl-mx--1 - nz-l. nx-- • .,. dZ"

ao-called "chain rule," or rule for differentiating The next result is the so-called y(x), eo 10 a function of a function. Informally stated, if "u - u(y) ,,(y) and y - I/(z), that u - "(,(21», u(,(x», then that" flu flu dy . - fly

'di"'

I a. IIIWf YALVII a....... YAL1lII ftJIOUK 'I'IIIIOUII

_ 111

" .......... 1M LeI U and _ V v ". PropoefdoA. 1M .". open ...".." ,..,.,. 01 R, 8, _ and ", let I: UU - V, ,: V -81M fwtdtou. /uftd,ioM. La U 1M . IUtA tIUJt 1'(St) 'eNe. V-R'" Let :roE %IE U". . IIuat I'(~ -and glU(s.» I'(J(~) 'aUt. T_ 0 I"(~ . .. . and _ T1&era (g o/)'(:ro) (g 0 J)'(s.) - I'U(St»/'(:ro). n'(~ "(J(~)/'(~.

For any fixed y. for which I'(Yo> existe, Nt

A ~

{

,(y) - ,(Yo> ,-~

.f y

•

,'

if y - fit.

E V, y ~ ~

Then

,<11) -

,(Yo) - A(y, ~)w

- ,.)

for all J/y E E V. Also lim A(y, y.) ... ,'~ - A(YI, y.),

.--

A{JI, ,,0) y.) is continuous at ,.. 80 that AW, fit. Now let Nt II. fl. "'/(~,Y -/(:ro),,, -/(z). -/(~). Since a oontinuoua function of a continuous function is continuous, continuoUl

...-..""

lim A (J(z),/(Zt» - glU(~)· I' (J(~). limAU(~),J(St»

.

Hence (g 0.f)'~ n'(s.) _ lim ,U(z» ,(j(~» - ,(J(:ro» _ lim A{j(z),/(~){j(z) A(J(z),!(:ro)} ([(z) -/(tt» - J(tt»

,(j(-» _....

....

z-_ z-:ro

....

z-Zt ~-:ro

-lim A(J(z),/(s.» • lim [(z) J(s.) - I'U(z.»/'~ gI(J(s.)}['(s.).. - lim AU(z),/(~) I(z) --/(~ •.

.... ....

.... .... z-_

~-St

II. ALtJE THEOREM. I S. TIlE THE MEAN V VALUE

""*'

ProptMldoA. La reol-tHJlued fwu:tiIm /1mdioft Oft Ot& , . opM of R 8 IIuat tIUJt ",...,.... LeI I be "a reakGl_ Gft. OJHIA aubaet U 01 CIUaUIa •a trlaimum ~ or " m___ cIllhe poitit _:eo E U. Then if 1 ia diDmnCIUGiu a minimum til t1&e point T1&era if/ diD".· IiGblI at til St, I' (:ee) ... O. liable Zt, I'(Zt) - o.

exiata a& real number' number a > 0 BUch If I'(~ I'(s.) "0, ~ 0, there exiate Buch that if z,. ~ '" Zt :ee and

Ztl < a Iz - Stl 3 then

I/(~)11-_

I.

II'(St) I /(Z) - /(:ee) I(z.) -I'(s.) -I'(~ < II'(~ z-:ro . 2 'I

that is,

/'(:ee) 1'(-)

II'(ZO) I < fez) [(~) -[(St) I'(Zo) + II'(~ IJ'(s.) I . I['(s.) - I(~ < I'(~ 2

11-_ z-:ro

2

104

v. V.

Dlrn:RIINTlATlON DIPTERIlNTIATlON

Since II'(Zo) II equals either I'(Zo) /'(Zo) or -1'(Zo), -/'(Zo), each of the two extreme terms neceBllarily either l'(z0)/2 of the lut last inequality is necessarily /,(z0>/2 or 3/'{Zo)/2, 3/'(Zo)/2, both of which have the same sign as /'(Zo). Thus if z '" ~ Zo and Ix Ia: --:ttl ul'{Zo). z.1 < 3,a, then - I (zo) ) I(z constant sign. But since I/ attains a maximum (J(x) -/(xo» /(x - xo) hu has a eonstant I(z.) is always nonpositive or always nonor a minimum at Zo, I(z) /(x) --/(xo) henrp, always negative or always positive if sx ,. :tt z. negative for all z E U, henre rol < 15. z -- Zo can be either and l:t Ix - xol 8. On the other hand the denominator % - z.1 < a positive or negative. Therefore we can find an x)l x ~ Zosuch that Ix 1% -:ttl 3 - I (ro) ) I(x - :to) and (f(z) (f(x) -/(xo»/(x xo) is either positive or negative, whichever we wish. This contradiction proves that the 888umption assumption that/'(Zo) that /'(Zo) ,. 0 is false. falee.

E H, a4 < b, be aCI continUDUa Lemma (Rolle'. theorem). Let a, bhER, h, and lell let/btl continuotU rt.al-t'alued fl'al-l'alued lunction junction on (a, la, bl that i. is differentiable Oft (a, b) h) and IUCA suc1& tAm tMt I(a) := j(b) := exists a number c E (a, b) h) IUCA suc1& tAm tMt /,(c) = I(b) = O. Then there erial. /'(c) - O. For since (a, la, bl is compact, compaet, 1 j must attain a maximum at at leut least one bl, and also a minimum. If both maximum and minimum are point of (a, la, b), points a, bh then since I(a) I(b) .. 0 we muat must have attained at the end pointe j(a) .. -/(b) j(x) .. % E (a, bl, I(x) == 0 for all all:t b), so sO that/,(c) = 0 for all 0c E (a, b). In the contrary case, Ij attains a maximum or a minimum at some point c E (a, b), and the previous result gives /,(0) j'(c) = O. A slight generalization of Rolle's theorem is the mean value theorem given below; Rolle's theorem is the special cue case where I(a) j(4) -/(b) - feb) .. O. The mean value theorem is illustrated in Figure 26.

FlOURIl FlOUR.

26. The geometric IIen!le ICnN! of the mean value theorem: the graph P'&Pb of •a dilf_&iabJe dift'erenUabIe Ieaat one tangent parallel to any chord. This Thie ie function has hu at leut is iUuatrat.ed i1IU1trated ICveraJ functionll. function •. The bottom curve ahoWl for !leveral show8 how the theorem falle faile K H dilTf'rentiability dift'l'rentiability is ia mill8ing milling at one point.

13.

_lie

DAN YALUII

'1'1IIIO'"

III

Theorem (Mean theorem). Let a, "b E R, b, _ and ,., [lIN (M4NUI _ue eIaeorem). R. a < ". ". •a jufld.ion on 011 (a, [a, b] tli6~ 011 continuoua real-t1alved r«Jl.«Jlued/uflditm b) III4t tMt ;. " di8~ OIl (a, .). 6). 2'AM TAM tMre ui'" m. a number e E (a, b) ItICA tMt IAere (a,") IUCI\ III4t [(") -[(a) - (b - a)f(e). I(b) -/(a) o)f(c). To prove thi" this, define a new function 11': F: [a,,,) [0. bJ - R by F(z) .. /(6) -/(a) - [(a) • (z - I(z) - I(a) - 1(11) (:I: _ .)

b-o "-a

is the vertical diatance for all z E [a, b). (Geometrically F(z) ia c:Iiatance ...... ~weeo the lIeIIllent throulh tbrouah theead 01 ... thiI If&ph of lover [a, b) bJ and the line lJeIlllent theerul poiat.l poinwof IJ'&Ph.) Then F II' is continuous on [a, graph.) ia continuoua (0, b), bJ, differentiable on (., (0, b), and '(0) 11'(") Rolle's theorem, there exiaw axist.l ace b) such F(b) - O. By Rolle', acE (a, II) luch that ~t 1"(0) "(e) - o. O. Thus Thua

rc..) -

"(e) [(b) -/(.) - [(0) -_ 0 F'(e) -_ I'(e) - 1(")

-a bb-o

'

proving provinl the result.

eorollmy 11 a retJkaluetl retJl-tNJlued [unction on em . opera ita R Au Corollary 1. II Itmdion [I 011 . irtltnal w.nal i1I _ivaliN ",,0 at each point, tIum tkrivotive tI&en I "ia COMIaftt. comtcmt.

"'0

We have to ,how show that for any points 0, a, IIb in the open interval we have I(a) .... 8Uppoae •a < b. For lOme e E E (a, 6) b) I(a} - [(b). I("}. Without 10M lea of generality suppose we have I(b) - I(a) ... .. (b - a)l'(c) a)l'(e) .. - O. Thus indeed [(a) I(a) -/(b).

CoroIImy 111I and real-tNJluedltmClioM opera irtltnal i1I in R Corollary J. 1/ anti fIg are real-t1aluetl jufld.iDM on 011 an . . iIWncIl ",AiM 1tdH tAe .me I and diD. 6rI by a COftIIanl. conet.cmI. tDAicI& Mve lite .. me tkrivotive _illative at eM! eaM point, lAM tIum / _ fIg tli6er For (J (f - fI)' g)' -

J' -

g' - 0, 10 80 I fI'

- ,fI is i. COIlItant. conatant.

DftIift.doft. A real-valued function Dft!in'tlon.

ir&cretuinrI) incretUing} *idly ineretUing iftereai,., ~ ~

*idly tkcreuing *idlfI tltft:retui,.,

011

a sublet. 8ubset U of R is ia called

if, whenever a, E U a,"b E and a < b, we have (and

J

{{ I(a) /(a) S/(b) SIC")

[(a) < 1(11) I(a) [("). [(II). I(a)

CoroUary I. 11 I;' 0a NIlIHHIlued ~juftJ:tion _. , . ..,.,.., lAst lot Corollary 1/I" IUf&CIioa .. OIl ... OJNI' inIfnal i1I ift. R IUe ... a poeitiN (...,.) (....,.) ....... ~ at .... . I/ " ;. tIrit:llJ ",..., ....., __• .., eocIa. poW, point, . tAea

(atridlr/ ~). (atrldlr ~). For if a < ", II, then/(b) --lea) /(a) .... (6 - a)/'(c) thesameaipu/'(c). - (b a)f'(e) baa thuamuip _/'(0).

U. TAYLOR'S TAYWRtS THEOREM. .t.

Let U ~ an open aubeet BUbaet of R, I: UU - R a ditJerentiable differentiable function. If function J': I': U - R ill lay that II ia is twice ttDice di6erentitJbk ditferentiabk the fundion ia differentiable, we .y call (f)' the teeond ~ derWaliH 01 I, writilll writinl (J')' (j/)' 81 88 J" / " or 1(2l. /(1) ill and can /(1). If 1<1) ia di'.-&iable, differentiable, we .y lay that I ill ia til,... tI&1w limu lima di6erentiGbk ditferenliabk and call (j(I», (j(I)' the ".,. (J(I)' 81/'" IAinI tlilrVatitle 01 I. I, writilll wriuns (j(l)' .. 1111 or 1(1). I"l. Similarly for functiona functioJl8 that are "4, 6, 6, • .• •• timee 5, 8, times differentiable. For any integer n > 1 and any St E U we.y that I ill lima di6em&tiGble ditferenliable CIt at Zt Zo if the restriction of I to Sf ia " Ii"", IOIIl8 open ball of 01 center Zt Zo ia (" - 1) timee times differentiable and (j(II-l)'(St) 101M (j(-ll)'(s.) eDat.; we then write (j(-l)'(s.) (J<-I)'(Sf) - p.)(Zt). eDt.; 1(·)(St). ThUll ThUB I ia is "n times differentiable, lor a liven given poeitive integer ", fa, il is "n timee times ditJerentiable differentiable for if and only if it ia each point of U. The ". ,," derivati,.e derivatiye 1(·) 1(11) of I ia is often denoted at IIch

en'

If.,

_~ r«.) -~ ...

NIIt of tbia leCtion l8Ction " will be one of the nonnegative nonneptive integera intesera In the nit 0, 1, 2, •••• .... For conyeaience convenience the . Nf'O'A /fmt:lirm I ia is defined 0,1,2, . . deriIH:IliN ~ 01 0G ~ to be IWI - f. I. Recall that

"I - 1 • 2 • 3 ... " ..• ,80 ,10 that if" - 1,2,3, ... (fa + 1)11)1 - (" (n + 1) • nl. (" "I. laat equation a1Io aIao hold if" In order that the lut if n - 0, we define 01 - 1.

lMn..... Lemma. l..c lAC U be open itt.tm1ol in i" lima di6erentiGbk. II lor (" + 1) limu

.,.,t'on

1M epotion

Oft CJft sf&tervGl R OM and kt kI 1M IUftClift/: It#&ClUm/: U - R be ditferentiabk. II CJftti 0, G, b E U '" _ tUJine ~1UJ R.(b, R ..(b, 0) G) E R bti b" Oft"

1(6) _ 1(0) I(G) + r(G)(b G) l(b) f(o)(b - 0) 11

+ I"(o)(b /"(G)(b -

+ I'II)(G)~ p·)(o)(b ,,1 d • R.(b, s) -

G)I 0)1

21

0)· .)"

+ ...

+ R.1b R.(b, 0) G), , ~,

t<,*l)(s)(b - s)· n'

&Il7 •~ E U we have For &Il1

l(b) 1(6) _/(~) _ I(s) + 1'(.) r(~) (b - s) z) 11

+ /"(:&) I" (:e) (b - S)I Z)I + ... 21

:t)1I + R.(b, .). ~). + P")(Z) p·)(s) (b :t)·

1'• .,..,.... • .. 'I'.U'O.' TAY14B'. TIIIDOUli

117

For fixed b, each term in tbia this equation is the value at s% of a real-valued function on U. Each of theee these functions except the laat is diIIerentiable, differentiable, hence alao the last. Differentiating Dift'erentiating both aidea sides of the equation we obtain

I'(s) + (/'(z) (I'(s) tl o -=_ /'(z) th tk

+ (J"(z)-!!(r(s) tI dx tk

s) (b - %) 11

Z)I (b - S)I

21

+ (p.l(z):Z (tAl(S)! +

_ I'(s) _/'(%)

+ I"(s) I"(z) (b - s) z»)) 11 I'''(z) (b - s)· Z)I ) + ... + /'''(s) 21 21

(b

:I

tz

~t)· + + J
R.(b, s) ~)

(-I'(s) + r(s) + (-1'(2:) /"(%) (b (II - s») Z») . 11

+ (-I"(S) :1:) + /"'(:1:) 2;)1) + ... (-r(s) (b - s) I"'(s) (b - S)I) 11 21 + (_/(a)(z) (_p.l(s) (b -

%)")

Z)_1 + Ic /(,.+I)(z) -!!-R (b s) z) s)-, ...."(s) (b - s).) + .!!.B.(b "1 dzk II ', ftl t

(ft - 1)I 1)1

....l)(s) (b -_/c /(ra+l)(z)

:t)· + !:z R.(b, s). z).

Theorem (Taylor'. theorem). 1M l~t U be "" an open inlerval interval in R anti and let the /tmditm tima tlifferentiabk. differentiable. Then Tlum lor Jor ""11 any a, b E U /tIfu:litm I: U --. - R be (n (ft + 1) lima

we"'"

wAaH

+

+

I(b) -!(G) /'(a) (b - G) + [j&(b I"(G) (b - G)I a)1 + + ... ··· -/(G) + .lJ!l.(b 11 21

/(")(a) (b _ + IC·)(G)

ftl nl

a)" + !(a+I)(c) 0)"+1 G). c....I'(e) (b -_ G).+I

.

Were ce illDfM ia tome number ftumber between G and a.ntl b (or, where (0',

(nft+l)1 + 1)1

'

if a anti and b are equal, c = == a).

Thi. 0, - .b, assume that G" a ,. b. We need to show that This is trivial if G b, 80 881ume !(,,+1) (c) Ic ....l)(e) R (b G) a\ .. (6 -- G) 0),.+1 R.(b, - (ta + 1) I (6 ....,

+

• ,1 (fa 1)1 IOIIle c e between a and b. Since G for lOme a ,. " b there is a unique real number K IUCh that such

(b ...., (b -;.) - a)"+1 B.(b, G) .. - K (n (ft + 1) R.(b,o) I)!I ·

+

The function I(J: rp: U .... - R defined by

(b - s)·+1 Z)·+1 rp(s) -== R.(b, s) "(:1:) z) - K (ft (n + 1) It

+

!p(a) -== rp(b) for all sz E U is differentiable. Furthemlor.e Furthernlo~ \9(a) ~(b) -a= O. o. Thus the restriction of f/J rp to the interval [a, b) (or to the interval [b, lb, oj aJ if 0a > b) Batissatis-

1.

V. DlrnJUlNTJAftON

fies Rolle'e theorem. Hence ,,'(c) .,'(c) - 0 for eome flee the conditione conditionl of Rolle'l lOme c between and b. Since

G

.,'(z) j
K (b + K:

s)s)· ftl tal

we have K - !(-+l)(c). /(rt+I'(C). Thus !(_+l) (c) fCrt+ll(c) R_(b,o) 1) I (b - a)"'I, 0)_+1, R.(b, a) - (" (ft 1)1

+

88 u

W88 was to be proved.

C888 "n -= 0 of thie thil theorem ill tbat the cue is eII8etltially eseentially the mean value Note that theorem. There ill is a little more generality here in that tbat it ill is not 88IUJIled U8UJned that 888Umption that! that I is il differentiable on an open interval conoa < II, b, but the 88lUmption taining is considerably tainilll 0G and b il coneiderably more etringent Itringent than the analogous condition COnditiOD in the mean value theorem, where/W88 &88umed differentiable only beCween where! wu 888umed ~ is not difficult to get a IOmewhat somewhat more long-winded ao and b. However it il IODI-winded ,tatement Taylor'l theorem which ill statement of Taylor's is an authentic pneralil&tion generali.at.ion of the (1188 subsequent lubeequent Problem 15). mean value theorem (see "Taylor" theorem" we have attached to the above reeult The term "Taylor's result iI milnomer. Taylors Taylor', original oriBinai ,tatement .. much weaker. is a convenient misnomer. statement w W88

PROBLEMS -+ R if 1. DiIIeuse Di8cu1l the ditTerentiability differentiability of the function J: R .....

(a) J(z) - {

{

Uin!

z It

ih"O

o ih-O (allUme the paeral paeraJ propertiel of the sine line function are known) J(:r) _ { (b) J(z)

{

(e) J(z) (c)

zlsin! jfz"O zllin ! ih" 0 I t· · z

o v'lil.

ih-O ifz-O

=

2. 1M / on the open subeet U of R be differentiable Let tbe the real-valued function J dUlerentiabie at e U. • the point _ E /(- -1). (a) Prove that I' /' (a:.)
u. .

(b)

t
PlIO.....

...,....... 3. Here is a "proof" uproof" of the chain rule:

119

,(J(-» ./(.) -/
,(fez»~ ,([(Se» _ lim(,{J(s» lim('(/(Z» - ,(J
(,I 0 I)' (%0) lim (,0/)' (%e) --lim

..... ...... .....

/(z) - f(z.) I(z) -/(z.)

»......

~..

a -_ z..

- ,'(I (Zo» 1'( /'(Zo). y(f(%e» ..).
AlBumlng the elementary properties of the tripnometrie . , tItM 6. Assuming trigonometric fUllC!tioDe, lul'letioDe, . IIaow daM tan zz - z is strictly increasing on (0. function lin z s ia is II&rictIJ atrict1r (0, 1r/2), while the functioD d~ng. zS decreasing.

8. Prove that a differentiable with IIouaded houDded dIrifttIYe derlfttI.. diflerentiable real-valued 'UDOtiOD fUDOtloD on R witb I, uniformly continuous. eontlnuoUl.

e.,

7. !At Let (It" G,. E R,.G< ., rea1-vaIued ...... caD b, and let I be a dllerentlable ...... valued ,fwaotloa OD . . opeI1 subaet aubeet of R' •. that eontaiDl eontaina IG, b). Show that if .,. II uy open (a. h). 'Y I, aD1 real_her nal1lUlIlber be.. I'(G) and /'(6) I'(b) then there exists exiate a Dumber number c (.,6) . . tIIM tween /'(G) (CI, 6) . Mlch that .,. '1 I'(e). (Hi",: (Hi"': Combine the mean value theorem with the lntermecllate /'(e). iDtermecllate .,.. value theorem for the function (J(Za) - sa> on OD the lit let 1~ f(s.. (J(z.) -/(Z'»/(Zl -/(~)/(zl-~ .. &'t)E.: • S ~ s. Zl < Zt %t S ~ 61.) bl.) G

e

-> e.:

Let.,. e .,

8. Let a, b E R, G < b, h, and let 1" I. , be continuous rea1-vaIued rea1-valued fuaetiou fUDetioJul on 1-, (-, II 6J (G, .). number ce (.,6) that are differentiable on (a, 6). Prove that &here there exiate exiata a DUIIlber (_,6) such that 1'(c)(,Cbr-,'(c){J(.) /'(c) (,(h)"=- ,'(c)(/(b) - I(G». I(a».

e

,(G» '(0»

(Hint: Consider the function

-,(G» g(a»(fe6) I(a».) ,(0» - (g(z) (,(z) - ,(a» (J(6) --/(&».)

(fez) --/(G»(,(.) F(z) - (/(z) /(0»(,(6) -

(Cauohy mean value theorem) to prove the folJowlna 9. u. Use Problem 8 (Cauchy followiDl ...... ..... L'Hoapital" rule: of L'Hospital's Ca) Let U be an open interval in K H and let I and II, be diflereat.iable difterentiable ............ ......... (a) funetiona on U. U, with, and,' nowhere 181'0 a be aa of functiona aero on U. Let Let,. IA extnIMll.r extreIDItlo1 Supp088 that limJ(z) - lim ,(z) - o. Thea U. Suppo8e

u.

-..... .... ,z.-.... .... ,o.

, lJ!l IJim W(z) _ I' lim tJ!l t:«Z}) Z - ~ yCz)

~ ,(z) if the rilhti-band right.-band limit uisU. uista.

&me .. &IIWD8d tbat (b) Same as (a), except that it is &8IUIIl8d that

. I' 1 I' .." 1 1· ,(:I) 11 - 0 0• :! ~ 1(:1) J(z) - :! ~ ,(z)

e.

e

(e) Same .. R·and •a ia &I (a), except that U - fa fz E R ::1> : s> a' al for IOID8 IOIne a eft-and II eymbol replaced by the symbol (el. Prob. 8, Chap. IV).

+.

118 u.

Y. DIJIDUNTIATION 1)I1'1'.......U'1'IOII

(d) Same .. &I (c), except that it is aaaumed 888umed that

lim _1_ 1 JIll lim' _1_ 1 .. 0 -_/(s) __ g(s) ....... J(~) - ...... g(z) - .•

10. State pnciaely preciaely and prove: An " times differentiable ditlerentiable fUDCtion function of an "n times dilerentiable tunes differentiable. dilerentiable. differentiable function is " tiDies 11. !At 011 aD open Open subeet ditlerLet I be •a real-valued function Oil I1llxlet U of R that is twice durer· eatiable E U. Show that if !'(Zo) /"(Zo) < 0 U"(~ U"(~) > 0) then entiable .t at %t StE I'(~ - 0 and I"(~ the restriction of I to lOme lOme opeD open ball of center St %e attaiDa attains a maximum (minimum) at %tats.. 12. A ... real-valued intMval in R is caned called COIIIIG conva if no point on ·valued function on an aD open interval the line ita II'&ph II"&ph lies below the lraph. graph. If UDe eepnent between any two points of its the function II this condition is kDOWD known to be equivalent to w the 18 diflerentiable, differentiable, thia COacntiOD tanpnt to the ao*ntioD that DO point of 01 the graph Ir&Ph lie below any point of any taDpDt If&ph. paph. If the function is twice differentiable the condition is known to be equivalent to the eecond derivative of the function heinl being nonneptive nonnegative at all pointe. State theee them. tbeee conditioDl coDditiODl in precise analytio terms and prove \hem. 18. abow that if 118 / is a real-valued function on aD an open OpeD IUlxlet subeet la. U. Problem 9(a) to Ihow Vof II It dilerentiable at the point So %0 E U U then U 01 R that 18 " tim. dilerenfiable lim n ~ ~

I(z. - /"(Zo) ~ I(St + II) A) -/(z.) - I(So) -/'(Zo) -I'(:co) !!! -I"(s.) 1r 2! 11

21

••. - I'-'/(:co) /<,,-l)(Zo) - ...

~ (a - 1) I, (n

Aa A" -

[<.)(it) [<"'(it) nl .·

Taylor's theorem to prove the "binomial theorem" for positive 14. U. Uee Taylor'1

iDte&ral inteFal

R: exponent n: ncaa-IS + "(fa "(,, -+ ftG_-IZ

1) a...-a,;a l)(n - 2) a.....aza o~ + n(n - 1)(" O,,-I%, + ... +;ca. + sIt. 2 2·3 15. Show that Taylor's theorem may be Itreqtbened 1&. strengthened .. as followa: follows: Let ut I[ be a& con· conreal·valued function on the olosed interval in R of extremities a0 and tinuous real-valued •II that is exIa (ft (" + 1) times ditterentiable differentiable on the open interval with these same ex· I1lPpoae that lim I'(s) , lim I"(s) , ... ,lim/III'(s} tremities and suppose lim/'(z), lim/"(z), , lim!(·)(z} exist and that

+

s)" _ 0a" (a + s)-

....

........ .........

1',/", ... .. .,/e-) ,1'''' are bounded. Then 1(11) -/(a) /(IJ) -/(0)

(lim/'(S»(II - a) + (lim/,(z»(b 0) + ... + (lim/III/(s»(11 (Um/(.)(z»(b - a)a a)_ 11 ._ "I ..... 11 .... al

+/(•• (c) + I'"+I/(C) 1)

lOme c between 0a and •• for lOme b.

(II (6 - a)II+1 a)"+1 i)!I (n

+ i)

CHAPTER VI

Riemann Integration

w. cIJIcUII ctiIcuII in thie thia

chapter the definition &lid uid buies buio propertiel of the Riemann intep'a1 intecral for real-valued funefunctiOD8 functiolll tiona of one real variable. The integration of functioDl eeveral real variablee variables will be diacUSBed diacWl8eCi in the laat of aeveral pointe of the onechapter, topther with some finer points variable cue. Here we are concerned only with the timplE8t reault., reeulte, up to the integrability of a continuous aimpleBt function and the fundamental theorem of calculUB. calculus. The details detaila of the proof. win will be the only eI8eIltially eeeentially new material for most atudentl. atudente. In the lut, laat section we apply living a riaoroua rigorous treatment of the our result. by IiviDI logarithmic and exponential functiona. functions. loprithmic

111

VI. RIEMANN JllllMANN INTlJORA110N INTIIOJlATlON

11. DEFINITIONS AND EXAMPLES.

DflJinition. Definition.

Let a, b E R, a < b. By a partition 0/ oj the cloaed closed interval

(a, %0, %1, Xl, ••• ,,XN [a, b] b) is meant a finite sequence of numbers Zo, %N such that a = = Xo Zo < Xl %1 < · . .. < XN = b. The width of this partition is defined to be

max {Xi (%,

-

Xi-I:: %i-I

== 1,2, ... , NJ. NI. i -=

BUm lor jor /J corruponding correaponding If J I is a real-valued function on (a, [a, b), by a Riemann aum to tA.e the ,iven given partition is meant a sum

f

I(x/){xe - XI-a), f/(X/)(Z4 Z'-.), 1-1 c-t

where %1-1 Xi-I :S S X/ x/

S x, for each ii-I, == 1, 2,0' 2, ...0,, N.

Zo, Xl, ••• ,J XN %N Thus, given any function J: [a, b) -. - R and a partition %e, lote of Riemann sums of [a, b), there are lots Bums for /J corresponding to this thi. partition, depending on the choice of Zl', %1', %1', cue where z.', ••• , XN'. In the special ease !(x) Bum can be considered an fez) ~ 0 for each x% E (a, [a, b], b), each Riemann sum approximation for the "area u area under the curve y = == I(x) between atJ and bit, bU, that is, the "area bounded by the x-axis, the graph of I, and the lines x == = a and x === b", as illustrated in Figure 27. Z1. However this geometric interpretation must not be overworked for at least two reasons. First of all we do not want to restrict ourselves to functions that are positive. po8itive. Second, arguments must have validity independent of geometric intuition. our argulnentB But the geometric interpretation does make the following definition reasonable. 0

•• ,

Dejin.ition. Dfflinition. Let a, b E H, R, a < b, and let J be a real-valued function on [a, b]. We say that J is Riemann integrable on [a, (a, (a, b] b) if there exists a number A E R such that, for any E > 0, there exists a 6a > 0 such that 18 -- A I < I!e whenever S 8 is 8.a Riemann sum for IJ corresponding to any partition of width less than 8. 6. In this case A is called the Riemann integral 01 [a, b) of \vidt.h oj bettoeen a and b and it isdenoted I(%)th;· J/ between J(z)th;.

f

makes 8cnse sense to speak of the Riemann intP.gral It rnakes in~ral of I between a CJ and b A, A' are Riemannintegr&1s since A is unique, by the usual argument: argulnent: If At Riemannintegrala between a and b then given any e > 0 there exists a 6., > 0 luch of fJ bet\veen such that IS - A I, 18 - A' I < Ef whenever 8S is a8 Riemann Rienlann sum for J corresponding [a. b) to any partition of la. h) of width less than 6. 8. There are partitions of [a, b) of widt.h 1f'M IPM than any pl'C!!l'ribed prescribed positive posit/ive number nUlllber since, for example, the partition by N equal subdivisions (with %i Xi = a + i(b - a)/N for i .. == 0,

,1. DanNnlOHI AND UAIO'L1II

•

./

./

7f

~

I

,

•I

I

~. S.

"-

r.

%,

1 1 1 1 1 1 1 1 1 1 1 I 1

1 1

,

, ,

1

1 1 1 I I

I I I

fAI' , s' I I

i

/

,, ,,,

1 1 1 1 1 I G

t;rz

"

1 1

/1

I

fAI' ' ,IS.'

fAI' ~t' , I

St

111

•,,I

It S.

I

. : II' 6, z/:.t,

~

I

s.

It

z, ~.

FIOUD 27. Area under a curve approximated by a Riemann IUIIl. lum. The IDdlcated iacUcated obolce oboioe Flouu ,;,',2:1', ••• ,t ,;,' IUm corNIpoDdinc corrwpoodi", ta of St', ZI', •.• z,' give\! Kives a certain Riemann eum to the ...... partition So, conaidered aD approxlmatioD 2:1, Sit fAIl, .' .' .. , " , ';" and thi. thil lum can be COJIIldered apJ)l'Olllma&ioD 01 the area under the curve. The maximum (minimum) value 01 of the 1UemaDD lUemaIua lum_ for I oorre.pondinl corre_pondlnl to the liven partition Ia liveD by the IQIIl IUIR 01. ~ the luml &real of the taUNt talleat C.bortelt) (Ihorteat) reetansl. redanp. 1ft III the 8Iure &pre 01 .... t...-.L let. ..J. areu [%1, :rtl, [ZI, z,l, •..• "., (z., [z., Sa), ,;.1, and the "true" "area under the Ute eurft" eurve" mun m_ lie betheae latter extremes, extremea, u doea our qnal Riemann aum. IUM, 'Olu ThUl the tween these as doee oriIinai Riemum eurve II ill error in making our original approximation to the area under the eune moat the total of the dift'erenet'8 dift'erenCftl in area between the tailed talleet and the at m08t IIhorteat rectangles. rectanglea. It seems aeem8 reasonable I'l'uonable that if we divide (0, (a, "1 6) iDto iDta more shortest widthp approaching aero, JefO, then aD all our RiemaDD Riemann IUIDI and more pieCftl pieeft' of width.IlUIDI eurve". (Of will approach a certain definite limit, the true "area UDder under the eune". course the only way to make thi. thil rigoroue rigoroul ia to U88 UIe thia or another aDOtber prooedure to define the notion "area under a curve". For •a epeciIc epecifie curve eurve the lat_ latter notion need not exist, just as u limitl limits do not alwaya exist.) exiat.)

z.,

or

or

or ....

or

a)IN, which is small if N is larp. large. Hence we can 1, ... , N) has width (b - a)/N, Riemann 8Uln sum 8 for If corresponding to a partition of [G, actually find a Rielnann (a, b) a, 80 that the two inequalities 18 - A I < .,e, 18 - A" A' I < • of width less than 3, •. Since e• was an arbitrary positive poeitive number we hold. Hence IA - A' I < 2 2e. must have fA IA -- A'I- 0, or A A - A'. A'. Note the use of ~ in [.' J(z)dz J(~)tk .. as a& "dummy variable"; we oould

[.t L'I (u)OO. equally well have written LJ(t)dt, or Lt

J(u)du. equally well have written J(t)dt, or We follow the usual convention of saying that I/ is, or is not, Riemann (a, b), and in the former cue case writing J(~)tk, J(z)dz, even if lis integrable on [a, J is a& function defined on a larger set than (a, [a, b), by implicitly replacillll replacing I by ita its restriction to [0, (a, b).

fL'

11" lU

VI. ~. BRMANK IIDJIAIfIf INftOIlATiON IIfftGBATlON

ExAMPLE 1. %E E [a, b). bl. Here we have I. J(x) I(x) == .. c, a constant, for all x any Riemann sum Bum

....f

f,j(2:i')(s. 2:Q) ... c(b - a). f/(x/)(s. - 4..) s.-a) - f, c(2:i c(:c. - Z;-I) Xi-a)., ... - C(ZN - x.) w w Since all Riemann luml Burna equal c(b - a) 0) we have IJ Riemann Rieillann integrable on (ca, j(z)dz -- c(b - a). [ca, 6), b), with L'/(s)tk ca).

....

L'

One of the principal results reaultl of this chapter will be that if

L'

I is con-

tinUOUl on (a, [ca, b) then L· I(s)tk exiltl, that is JI is Riemann integrable on UDuoua j(z)dz exists, continuous [a, b]; b); this is trivially illustrated illUitrated in Exaolple Example 1. But if I is not continuoUi I(s)tk mayor may not exist, as &8 is shown by the following examples. then L· J(z)dz

L'

EXAMPLB ExAIiPLII

2.

Let

[CI, b), let c E E 8, aet Ebe a fixed point of [a, H, and let

I(~) I(x) _

{Oells-E. ~f s~ ,. E c If ~ - E.

For any Rienl&nD (a, b) b] 01 of width lell leu Riemann IUDl BUm 8 correeponding correaponding to a partition of (ca, tban ~ may be than • we have 181 < 21cl' 21cl' (the coefficient 22 appearing since E one of 01 the partition pointe pointl s. ODe ~. and we may in this cue have both s/ x/ and Zt+l' %t+l'

equal to c.) So 80 clearly

1

J:r. I(s)tk - o.O. j(z)dz -

, , - - ...... ., ------.......I

I

I I I I

•

•,.

•

l'Iavu.. lunotiOll 01 l'Iau.. IS. Graph of the function of Example 3.

ElwIPLII 3. Let Let. CI, CI, {J fJ E (a, [a, b) EnIlPLB b] withCl with a

< {J. fJ. Let/: - H Let I: [a, (a, b) b)-+ R be defined

(CI, If) II) { II if sx E (a, I(x) - { 0 if zx E (a, [a, b), z x (! (CI, I(z) (a, fJ).

e

11. 11.

lIS 115

DI:nNITlON8 AND IlXAUPLIIS BXAMPLES DllnNlTlON8

Let %0, [a, b) b] of width less than 3a and consider :ro, %1, ••• , %N be &a partition of (a, a Rienlann Riemann sum for If corresponding to this partition, say

t/

(xl) (Xi - X'-.), X'-I), 8 - fl(%a')(x, :1m

i-I 1-1

where :1'-1 x/ S ~ x, for i-I, 2, ... , N. Sincel(xl) SinceJ(x/) is 1 or 0 according as X'-I S :S %/ the point ~' is in the open interval (a, (a,~) ~) or not, we have

8S -

1:. L* (%a (~ -- Z'-I), Xi-I),

the asterisk indicating that we include in the sum BUill only those i for which xl E (a, (j). fronl among &lllong 1,2, 1, 2, ... , N such that ~). Now choose p, q from ~ a Xq-l < {j 5 XV' Xq• S a < XI" x., Xq-I ~ S if p + 1 SiS ~ i ~ q - 1 and ~: xi' ft fl:. (a,~) if i < p or i > q. Xp-l %.....1

%a' E (a,~) Then xl Therefore

1: L

(x, ~I) S (%i -- ~'-l) 5SS ~

~1S;':Sq-1 ,*1~iSt-l

1: I:

(x; X'-I). (Xi - %'-1)'

pS;S, .. SiS,

+

By the choice of p and q, (q - p + 1)3, 1)8,80 g, ~ - a S :S X, x" -- XXp-l .....I < (g so that if 3a is sufficiently lllust have p lufficiently 81nall Illlall we lUust 1 S gq - 1, in which case the last Bimplifies to . inequality simplifies

+

Z,_I Xt-l --

x,. x" - X,._I. X" S ~8 S S ~ z, Xp-l.

Therefore (Xt:-I - ~ (%p -- a) :S (X~I fJ) - (x. ~ S - (~ - a)

S ..... I - a). :s; (x, -~) - (j) - (X (%.-1

less than 3, Since the partition has width leas a, each of the quantities Xv-I z,.p -- a, XI z" -- ~, ZXp-l .....l - a a is of absolute value less than 6. X a. Therefore

-

~, {j,

/8 ~ - a)l f 8 - (~ a) I < 23. 26. Since a 3 W8B was an arbitrarily small positive number nunlber we conclude that I is Riemann integrable on [a, b) and that j(z)th; = fJ I(z)d:r; ~ - a.

J:

ExAMPLE 4. Define J: I: (a, [a, b) - R by setting J(x) I(x) = 1 if xz is rational, EXAMPLE b)--+ t.he restriction rest.rict.ion to [a, b) of Example /(x) .. == O. (This is the Exanlple 6. 6, page otherwise f(z) in R is known to contain bot.h 70.) Any interval ill both points poiuUI that are rational points that t.hat. are not.. and pointe not. Hence for any &uy partition .1:0, xo, ~I, Xl, ••• , ~N XN of [a, (a, bJ xl'l to be either all rational, or all not, in which case we can choose the x/'s BUms are respectively b - a and O. the Riemann Bums o. That is, b - a and 0 BUms for f corresponding to any partition of [a, b), are Riemann sums bl, 110 matter Dlatter what the width. It is clear that f is not Riemann Rielnann integrable 011 [a, b). b].

In the future, for: fo~ brevity, we shall say that a fUllction function is integrable on a closed interval, rather than Riemann integrable, and speak of its inUtlral iDltead instead of ita its Riemann integral. It should be borne iftUgral bome ill miud nlilld however processes than that of Riemann, that there are other integration processC8 Rienlallll, and for

116

VI. JURMANN INTBOB4T10N

theae other integration procel!l8e& procet!l8es our results resu1ts mayor may not be true. For theee example the moat commonly used integral after that of Riemann is that of Lebesgue. A given real-valued function on [a, b) bl mayor may not be Lebesgue integrable. If it is then its Lebesgue integral is a certain real number. If a function is Riemann integrable then it is also Lebesgue integrable and the two integrals are the same (hence can be denoted by j(x)dz). But many functions that are not Riemann the same symbol J.b f(x)dz). integrable are Lebesgue integrable, so the I..,ebesgue Lebesgue integral can be of Example 4:4 above is Lebesgue greater use. For example, the function of Exanlple as a matter of fact its Lebesgue integral is zero, in line with the integrable; 88 fact that in some sense the points of the interval (a, [a, b) bl that are rational are relatively few in comparison with those that are not. We repeat for emphasis means Riemann integrable, integral means that from now on integrable Dleans Riemann integral.

L'

,2. II. LINEARITY AND ORDER PROPERTIES OF THE INTEGRAL.

Propoa.don. integration hal ha. the foUowiftf jollowing propertia: Prop08ition.. Riemann inUgrol.ion Ij Ij and 9g are integrabk real-valued real-rlalued lunptiOM junptiona on tile the interVGl iflterllGl [a, [4, b] b) (1) If Ihen integrabk on [a, b) m&tl 400 ~ Ij + , it inUgrable (a, b]

f

(f(x) (J(x)

j(x)dz + L' I.. g(:z)dx. g(z)dz. + g(x»dz - fJ.b f(x)dz

(I) 1/ Ij /juan real-valued Junction junction on the intervCJl [a, [4, b) and (') is an integrabk integrable real-valU«l tAe iftlmJal CIftd R then c/ cj i, i. integrable Oft on (a, [a, b) bl and 400 cE R cj(z)dz - c f j(z)dz. J.' cf(x)dz L' f(x)dx.

These facts are easily proved by looking at the various Riemann sums, 88 8.8 follows. Given any E > 0 there are numbers ii, 810 8t ~ > 0 such that if 8 B1, 8S22 are any Riemann sums for J, j, (J9 respectively corresponding to parti, 6t rMpectively, tions of la, (a, b) bl of widths less than 81,6. respectively, then

IS1 f(x)dzl <~, lSI - J.' !(x)dzl

St 1Is,

f g(x)dzl < ~.

jf %0, XN is any partition of (tI, of width I.. le11 than Hence if xo, Xl, Xl, ••• ••• ,,XN [a, b) or min I(8ai, luch that %4-1 S %4' zl S Xl', ••• .•• ,,%N' XN' are such x4-15 5~ %4 for i-I, i - 1, 1, 6.1J and if Xl', ... , N, then

as

• a a,

(/(%4') + g(:zl»(:zo I ~ (f(:zo')

g(x/)(xi - %4-1) -

~ . . I( ~f(:ZO')(x, ... 1(

!(%4')(Xi - %>-1) X4-1) -

(I.' f(x)dz !(x)dz + L' I.. g(z)dz)1 (f g(x)dz) I

f !(X)dz) f(x)dz)

t.

+ ( ~ g(:zo')(:ZO g(xl)(%4 +(

%4-1) 4-.)

9(z)dz) I L'Lt g(x)dx)

'2. ,2.

I

s ~j(Zl)(Zi ~/(zl)(~ - Zi-.} ~-a) -

UN-.uJn An AND 08.,.. OaD8 PBOPIIII'IUII PROP..". UICSABI'IT

f I(Z)dzl j(Z)1&1 +I IEFa ,,(~')(i, g(Zi')(i, - zz,-.) ....) -

111

r. I /.' ,(z)dz ,(z)u

• • e• <'2+"2<'2+2--·· e

e

This proves part (1). For part (2), given any. > 0 there Ie. number' > 0 8u(,h IUlll 8 S for /J correepondilll correapondinl to any ,.ntllon partition 01 of sut,h that for any Rien18nn Riemann sum

18 IS - J.'

(a, J.'/(z)1&1 t/lel (it iIill permis....... (a, b) of width less leu than 6 we h&ve J(z)dzl < ./lel sible to restrict our attention to the cue e c .. pi 0 if we note tbU that the ... __ c :Ill Xl, ... ••• , , %11 of ,fI,'} 'fI, II] 01 of wid'" wlcItil - 0 is a triviality). Then if Zo, x., XN is any partition 01 xl E (Xi-I, (Zi-l, Xi] Zi) for i ... less than 86 and x/ =- 1, ... ..• , N we have

I~t.1 cJ(~')(~

L'

Zi-t) - ec/.' I(z)dz c/(Zi')(Zi - Zi-J) j(z)1&

-lei'

I

1 It1t./(~,)C~

(Zi')(Zi --...} - -..s) -

m·.,

/.tL'/(Z~ J(z~ < 'leI· .1· Tor ...

finishing the proof. An imnlediate propolition iI it tbM that C (u.... immediate consequence conaequence of the propoeiw. ..... the hypotheses hypotheeee of part (1» (1»

f.'/.' (f(z) (/(z) -

f

g(z»dz - /.' I(z)dz j(z)U "Cz»dz

I.'J.' ,(z,.. ,(z)6.

This cornel fronl and from applying applyinl part (1) to the functional rUlHltiona J.... being integrable by part (2), with cc" - -1. beinl

-'t

latter -" the IatW

Propoaition. lin integrabZ. reol-NltMltl ~ /vItditm "" 011 1M ..""., ."",., fa, Ie, ij 6) ProJHMition. 1// 11 I i, ill an int""rtJbl. real-tIaluetl ~ O/or all Zx E (a, b), 0 Jor aU bl, then tlam

aM J(x) oM fez)

J.'f I(z)dz j(z)1& t! ~ O. any.e > 0 we may find a• Riem&lUl J en For if we are given allY Riem&ml BUm awn S 8 'or len

18 j(z)dz I< eo Then ."r. J: J(z)dz ICz)u ~ B --.. 0euIJ lS - L'J.' I(z)dz eo a.dJ' J. I(x)dz t! -eo Thia heiDI that L' j(z)1& ~ heiDi true Vue for .u • t > 0,' we ...,. ....

(a, (a. b) h) such that

8S ~ 0,

80

-f,

t! O. J:f./(z)dz /(z)1& ~

Corollary I. h) fWl/(z) oM /(z) (a, h]

For

f,

i"""

" .. .

II 10M" 111 MId , are i,.,.",..,. ~ NtJl.4HJl.., ~ .... ."". ......., ......, ,,(z) lor all sz E (a, I•• lilt b). Uum lAM S ,(z)

I.'Lt ,(z)dz -

L· .ez)dz. L'Lt/(z)dz j(z)1& S ~ l' fCz)dz. L· (,(z) - /(z»u L'L·I(z)dz" j(z)dz .... L' /(z»1& ~ O. o.

III lU

ft. Yr. BlJUUlfN IDIIIIAIIN INftOBAftON INTJIOIl&'l'I0N

Corollary an integrable real-vGlued real-valued function on the interval (a, h) b) eorou.". J. 1// ia ill (1ft and 1ft, J(z) S M lor for aU all zx E (a, la, h), b], then m, MER are such lUCIa that m S /(z) m(b m(h - 0) a)

~ S

f /(z)dx ~ M(h M(b - a). 0). /(z)tk S

For /.' rna s /.' J(%)dx S /.' Mtk, Mdx, and we know that for any conmcb:S /"/(*)tk:S .tant .taDt c we have /.'

=-=

c(b - 0). a).

I S. EXISTENCE OF THE INTEGRAL INTEGRAL. Lemme I on the ifllcrvol int-ervol (a, la, h) b) ill i. integrable on Lemma I. A reaZ-flalued reoJ.walued lunctioft /unction 1 (a, b) given afty any e > 0, there aUt. ezilt, G ftumber number 8 > 0 lUCIa auch thtlt that h) if tmd and only iI, if, ,wen 18 and St are arB Riemann ",m. aumB lor for /f correaponding corrupond·ing to UJ lB.1 -- 8Btlt l < e whenever tDltenever 8B.1 aM JXlrtitiona lea. than a. partitiou 01 0/ [a, [G, b) 01 0/ tDidth tDidt1t leBa them 8. Firat Oil (a, b]. Then given any eE > 0 there is a First suppose /J integrable on h). Theil lOch that •, > 0 luch

IB - /.'/"/(z)tk 18 /(%)dx I< 1/2 whenever S is a Riemann 8um for /

I. . than 8. 6. If B. 8 1 and SI St are corresponding to a partition of (a, b) of width leas two .uch luch Riemann BUms lOmB then

I

18. /(z)dx) - (SI (St - J.' f 1(:I:}tk) /(z)dx) I lB. - 8tl-1 BII-I (8. (S. - f.' J.' I(z)tk) SiS. /(z)dx I + 1St - J.' f 1/(z)dx < i + ~~ sa I.e. SiB. -- f.'J.' I<*}tkl (:I:)tk I <~ ThiI proves the "only if" part 01 This of the lemma. Convertely, Ulume &l8ume the hypothesis 01 of the ceif" uil" part of the lemma. For Convenely, 1,2, 3•... ft - 1, 2, 3, ... choose any partition of (a, b) of width 1e88 lees than l/n and a lum 8<") 8(11) for / corresponding to this partition. Then 8(1), Riemann 8um 8(1), B(I), 8(1), BCI), ••• it ie •a Cauchy aequence of real numben. (l4'or, asaulnption, for sequence 01 (l~or, by &l8umption, a. 8 > 0 lOch lB.1 -- Bt such that 18 St I < te whenever B. 8 1 and every •t > 0 there exists ... lor 1 corresponding to partitiOnl 8Bt1 are Riemann 101M IUIIlI for / correapondinl partitions of (a, b) of width 1_ than., intepr N lOch 3, 10 80 if we choose an integer 8uch that liN < a then we have Jell than IB(II) - SCM) 8(->1I < t whenever ft, 18 N.) Since R is complete, the sequence 8(1),8(1),8<1), ita limit be A. Given any I> E > 0 now B(I), Set,. 8(1), ••• ••• converges. convel'les. Let its 181 -- 8Btt Il < 1/2 e/2 whenever 8. 8 1 and 88.t are Riemallll Riemann choose .,• > 0 such that 18. 101M for 1 partitions of (a, b) of width lC88 sunlS J corresponding to partitiolls less than "8, and choose an integer intepr N 8uch Buch that 18(N) IS(N) -- A I < ./2 E/2 and 1/N < •. a. Then for any Riemann 8um Bum S 8 for / correeponding corresponding to &a partitiQ" &Il7 part.itiQ" of [G, (a, bl b) of width ... than I• we have 1_

B''', ...

AI-I(B 18 - AI-I(S

8(N»

+ (S(N) (8(N) -

'nwa Ie integrable intepable on [0, la, b). n ../J ia

A)I S IB - 8(NlI SiS S(N) I + 18(N) ISCN) - AI e e ~2+2-e. ~2+2==E.

18. I a.

UIftIIN. . O. IIlD8'I'IIMca 01' 'l'IIJI TIIJI INftOlI4L lNTS0a4L

119

At it would be easy to UI8 Lemma 1 to give a direct proof A, tbie point i' result of this section, eection, the integrability of continuous functions. of the main reeult However we postpone this reeult result because it is natural to wonder what integrable functions are like in general, and the next lemma and proposition und81'lltanding of the situation. will give us a better intuitive understanding D.tIIlddon. ~"'do". A real-valued function I on the interval (a, [a, b) h) is called a . , /wttJCioft ~ if there exist. ~ ~I, exittl a partition Zt, ZI, ••• ••• ,, ~" ~N of (a, [a, b) h) such that I is conatant on each open subinterval (2:0, zs), (~l, (ZI,:c,), (:r:N_I, SN). ZN). CODltant (~~a), St), .•• , (Z"_I, functloDi of Examples 1, 2, and 3 of ,111 are step funeFor example the functiona 1,2, tiona. A step function of more general appearance is indicated in Figure 29.

, I I I I

II.........., ,..--, I I I

,

I II , ,I I

I I II

I I ~ I ......, II .--. •t I I

,I I

r--t r---t

I

I I 'I I I

I

I I I,

I

,

FrOUD 20. Graph 01 a .... .tep IIJ. PlQU.... p fuoctioo OIl (G, la, ".

.A., A.,

£em.... ~ it irttegrable. inUf/rtJble. In particular, particul4r, il ~ Zit, ZI, ZN Lem.... J. fvrt,cnon Za, ••. ••• , ~N interval (., Cl, ••• , CN E R and if R .iI it •0 partiUtM partitiOft 01"'01 1M intmJGl (0, b), hI, if CI, il I: (a, b) hI -. -+ Ria aucA 1IttIl/(~) z..-l < Z . - IItGt I(~) - Co if if:CO-I ~ < ~ lor i-I, i ... 1, ... , N, tIum. tJum ·/(z)(#.. tw Co(~ - :t J./.•'/(s)dz •

:co-s}. a:.-a).

i0oi

Note thatthe 1(~,/(zs}, .. • ,J(ZN) eft'eot on the tbatthe valu. values of ol/(~,/(~I)' ,/(~,,) have no effect

intearal. intepal. It is convenient to place this lemma lenllna here, but ita proof could havo have

,1.,1

been given much earlier, immediately after the definition of integral in ill simplo to base the proof on 011 Examples 2 and 3 of '1. However it is mOlt most simple

t 2, as &II follows: '1 and the first proposition of '2, follo\\"II:

110

VI. VI, RIEMANN IUBIlANN INftORATJON IIITB01U.TJON

For ii-I, - I , 2, ... tpl: (a, b) -+ R by ' ," , N define 1('1: bl-

() - {I{ 0I

tp~ % ~z) Then J / -

tw

-

if Z (~I' iii) % E (%4-1, %() if z E (a, b), s~ e (4.1, ~). bl, z (~l, z.).

is a function on [a, b) that takas takes on the value aero at all .. points ~ Zl, :el, ••• ••• ,, ZN; ~N; hence it is the IUDl BUm of a pointe except poIBibly pcaibly the pointe :eo, '1. By Example finite Anita number of functioDB functiollll of the type of Example 2 of ,1. 2.,tolether 2, topther with the linearity of the integral, we have C4fP4 CIfP4

W

L' V(s) V(Z) -

~ CIfP4(z»a C4fP4(s»)a ..- o.O.

~ is a function of the type of Example 3 of ,1, But each "" '1, hence is i. intainte~ble, fPi ... :Ii Xi - 4-1. gn,bie, with ~ ~I' Again using the linearity of the integral we

L' ... get 1J ... VV- t ~) + t ~ integrable and t CIfP4(z»)a Ct'fP4(Z)~- + !r t CiL' ,J(~)dz -L'(/(s) L'/(e)d:e - L'V(z) - !r c, L' .,(s). .,.(:e)d:e 1 r 4-'--= t ~(~ - ~). 4-1). C4fP4)

CifPi

W f-I

•

(001 '-1

•

i-I

c,(1Ii -

'

4-l W

For an illustration of the situation of the following propoeition, proposition, where a functioD8, see Bee Figure FilUre 'n 27 (pap 113). function is sandwiched between two step functiollll, Proposition.. function If on t1&e the inIm1tJl intmJal (a, b) u iI intfJgrabltl iNegrtJbr. 'ropoaidoR. The TIM real-volued real-wluetl IUMUm each ef > 0, tAer, exist "", at" /1mdioM /uncliona II, 11, /. /1 on [IJ, (4, b) on [0, (a, b) if il ond tmd only onlrI if, lor eacA tAm eNt auc:1& tMt tIuJt aueh

Oft,

!t(z) I(z) S I,(z) /1(%) S ~ I(x) II (:r;) lor «ICIa eacA z % E (a, b)

and

L' V.(z) - /1(Z»d:e < eo

We first prove that if the given condition holds then IJ is integrable. Given e > 0 we have to produce a' a >0 We use the criterion of Lemma 1. Given. 8uch that if Bio 8B,2 are R.ienlann Riemann 8UIllS 8UIll8 for IJ corresponding to partitiollll such jf 8., partitioDl of less than than'B t.hen 181 - B,I (a, b) of width lese 8 , 1< •. t. Use the hypoth.s hypothelia to find functiollll/ I, on (0, [a, b) such that step functions 11,l , lion

a

II(z) 11(:r;)

S 1(,;) /(z) S I,(z) [a, b) I,(:r;) for ail all z:r; E (a,

and

L' V,(z) - II(s»tk < ;.

,3.

DJft'SIfCII OP 'l'IIJI .........

111

Since "". Itt I. are integrable on (a, Ul7 Rieme. [IJ, bJ b) we can and find a ,• > 0 auch that. that ..., mann IUm sum for II " (or I~ /~ colTelpondinl colTMpondinl to any partition 01 of (a, (0, ") b) of wicI~ wid~ 1_ ditYel'8 in in abIolute abeolute value from Lt Il(s)a ,,(z). (or 1aC..)d:e) by 1_ than ,• differa It(s)a) b.r Ie. than ./3. t/3. Now let 8 be any Riemann sum IUm for' for I colTMpOndiDI panttion correepondiq to •& partition of [4, Sit •ia tbiI panttion and [/I, h) b) of width lees Ieee than '; sa1 say that ., Zt, SI, ZI, ••• , ZII tbia partition

L'J..

J.'

that 8 that.

ft

i-I '-I

'i

,(%l)(%4 Zi-I), where 4-l Zi-I ~ %I' %4' I(ZI') (%I - %4-~,

Z4 for each i-I, .•. .. ", N. N, ~ Si

Then since

I(:c) S ~ /I(S) I.(:c) for :I:c E (a, (/I, 6] b) S J(s)

II(z) JI(S)

we have

t

1:/1(%1')(%1 I,(:c.')(zi - %4-1) ~,) S~ 8 ~ S ;..a ...1

t

tIt(Si')(S,-~ /t(Z4')(Z4 - ~.).. '-I W

By our choice of , we have

II 6 L'/I(s)U1 < t 6"(%4')(%4 -~.) - J.' /1(S,')(S,- Sf-&) -

11(:c)d:e1

and

implying implyinl

•/I(s)#b - T;e < 8 < I..J.'• 1aC.:c'" I.(a)a + Te ;.f;: I.J.'• II(:C).

Thus 8 belonp to a fixed open interval

(Lt/l(s)U L'/.<s)U (J.' 11(:c)d:e -- ;, ;, J.' I.(:c). + ;) of length oflengtb

1."1 I'll

()) . • .• .• .•

e 3+ • UI(S) -/l(S) .+'8 <'3+3'+.-" f T+.~OO-"~·+T 0 and we bave·to have to pmdaaae pJOduee ltep It, I. with the der.ired Uei. Lem. . 1, 'we . . step functions "". deeired propertia Uai. Lemma l,we CIlIa IUCh that aDJ' find •a partition ~ :elf z" ••• ' •. ,J _ ZII. of (a, (/I, II) b)euch &OJ' two RiemaDD Riemum . _ correepondinl to tbil this partition cliff. differ in abIoIute abiIoIute value by 1_ tbaa .. for / eorrespondilll 1. . tIIaa ", where , is thaD .. whenl .. ia lOme arbitrary fixed poIitive number Ie. than ~ That II, ill, for arbitrary (~l, Sil, arbitrar,y sl, %4', ~/ %4" E [Zi-Io %4), i-I, ... ., .,, N. we have

'*'

I6 ~

w.

(j(s.') (J(%4') -/(s,"»(s. - 1(:cl'»(Z4 -- Sf-&) ZI-..)

I1<<

i. e'.

111

ft.........

IlftWGItA.ftOK

tf lpecial cue where, for lOme fixed index tI we apply tbiI tbia inequality to the special i-I, ..., =-~" if i "i pSi and Zj" %/' - Zj, Z/, we get ... , N, we have ZI' alal' if. lID

I (f(z/) -

I(Z/»(z/ - s/_~ 1< e',

S, !,

impl:rlnl implJinl

1/(./) I/~l) I < ~I _

SI-I

+ I/(z/) I/(s/) I· I.

Thia 181' Ian inequality hola 1bia holdl for lor all Zj' SI' E I~ lSI-a, Zj). SI). Thua ThUll I ia is bounded on [Zl-I, (a, b). Thua Thus for. for i -.. 1, ... , N we lZi-I' %I). Zj). Therefore I iI ia bounded on all aU of 1.,6). I, .,., can define cande6ne • .I.b. lI(sl) I/(zl) : %i' (Zi-l, ~JI %4) I me - ,.I.b. ~' E [z....,

ttl( -

and

M4 -l.u.b. (f(z,') : ~' ~'E (4-1, ~JI ~)I M, -1.u,b,II(~') e Iz...., and .e ltep functions on (., (0, 6) b) by we can define atap functiona II, I., I. lion !I(Z) _ {~

if s.-l ~. < z - I, ... , N Z < ~" ~, i-I, if sz - ~" ~, i - 0, 1, I, ,... .. , N

IDln Iml, •••• til. I

/.(z) _

{Mt max IM

I , ••• ,

ifZ4-l<s<~,i-l, if %1-1 < z < %I, i-I, ... ,N ,N Zi, i - 0, 1, ,.•. if s - ~, , ", N.

JINI

II(s) S ~ I(z) S ~ II(s) I.(s) for all sz e Clearly Il(~) E [., (a, 6), bl, and the proof will be CORlconlplete if we can .how that /.' (f1(S) CI.(s) -/,(s»a - II(s»cb < .. To do tbia, for any number "tt > 0 find specific .pacific ~',~" ~), ,i ... I, ... , N, 10 that real Dumber Zi',~" E (Z4-I, (Zi-l, sil, =- 1,

L'

I(zl) I(~,)

e

<~ me + 'I, tt,

!(ZI") I(~'')

> M, -

'I. ".

Then

••1: E U(S4") U(~") --/(Zt/»(~ I(~,»(~ -

·N

w

Since

~.> ~a>

• N

E (M, > 1: -a ...a

tnt - 2,,)(~ 2")(%4 - ~-a> Zi-t> '"'

- L'J.' U.(s) - I.(z»cb - 2,,(b - .), (f1(S) -/,(s»a CI).

I~ U(~') -/(~"»(~ - I

Z4-a> < ,

we have webave

..

J.' (f1(S) -/,(s»a - 2,,(b -

CI)

< .',

or

L' Uh) - 11~»cb < , + 2,,(b - .),

14.

J11llIDAldNTAL TllmUM TImOUII 01' CALCULVS CALCULV8 , ' - rutfDAdl!lTAL

lIS US

Since" WI8 nunlber, we have wu 4ftl/ ony positive number,

L'/.' (JI(Z) /1(Z»tk S .' E' < "E, (J.(~) - Il(~»"

and the proof is complete. The fonowing coune of the proof of the foUowinc reeult, reeuit, which occurred in the COUl'8e proposition, is COll88Quence of the propoaition proposition itself. ia a trivial conaequence

Corollary. II the junction /I 07& on (a, b] ill is integrable 07& on la, [a, b), 1M real-valued NtJl..tHJ11Ultl /unction tAm it i. [a, b]. iB bounded on 07& (a, h].

Theorem. function on the interval la, [a, bl b) TluJorem. 1/ II /I iI iB a continuoua continU0U8 real-valued lunction tAm /.' /(z)tk I(z)tk eziaU. e:eiBU.

L'

We ahall prove this theorenl theorelll by showing that the criterion of the praoedinc proposition obtains. Since I/ ill preuedilll is wliformJy uniforluly (."OutinuoUB continuous on (a, bl, b), liven call find a& ,I > 0 such 8uch that whenever ~', z', ~" z" E [a, II) b) given any. any e > 0 we can ~'I < 8 'then I/(z') -j(z") - l(z")11< < e/{b _/(b - a). Chooee and I~' - %"1 then 1/(:1:') Chooae any auy partition Zo, bl of width less than ,. a. {I'or 14'or each i :; Zt, %1, ~l, ••• , %It s" of (a, b] = 1, ... , N choose zl, Zi" E (Zl-l, Xi) .such that the restriction of I/ to (4-1, (Xi-I, Zi] Zi) attaillB attail18 a mini#&e', #&e" (~""l' s;) mum at zl z/'. Define step funCtiOllB functiOI18/l 12 on (a, [a, b] by #&e' and a maximUDl maximum at #&e". II. Is

':c' -

t

/1(Z) Il(~) _ J/«~.)l) J/«~)')

11 .,..,

:c < #&e, ~,ii ... == 1, •.. ... , N if %i-l Z ....l < s if z .. . · • ,J N. - %i, Zi, i --= 0, 1, ..•

/.(s) _ J/«~.)l') JI«~i)'') if 4-1 z....l < Z < Zi, i-I, I.(z) i - I , ... .. · , N 11 Zi, i .. 0, 1, .•. ,N. , N. 11...., if z~ - #&e,i - 0,1, Il(z) S /(~) I(z) S /I{X) I.(s) for all s% E (a, b]. Then /,(%) b). Furthermore Furtherl110re for each i = 1, l#&e' - zrl x, -Xi-l - X,-l <', ... , N we have ,x/ xl' I S ~ Xi < a, 110 tW that If(x.') I/(x.') - f(oI:,")I J(x/') 1 < _/(b - a) and therefore /.(x) f.(%) - !1(X) fl(x) ./(b fore

< ./(b e/(b -

a) for all x E [a, b]. b). There-

/.' (J.(z) --/l(X»tk /&(z»tk S max 1I.(z) I/.(z) - /&(z) : z;l; E (a, la, b1l bll • (b - a)

< <-b b e_ •• (b (b -a

-- a) a) == == •• E.

1'hua the criterion of the last Jut proposition is satisfied. 1'hus I". THE FUNDAMENTAL THEOREM 14. THEOREl\1 OF 0 .... CALCULUS.

Prop08"I0",. I'ropo8itwn. Let a, b, c E ft, H, a < h b < c, and let f/ be a real-valued /unction function [a, c). Tium TAen J / i. 07& (a, [a, c) i/ (a, ia integrable on ij and 07&111 only if it ill is integrable on both both. [b, el, e], in VJhich umich ClUe caN [a, b) and lb,

on Oft

J.'

/.'/(z)tk I(z)"

+ J.. /'·/(z)tk. L" /(~)tk I(z)" - J." I(z)rb.

114 1M

VI. VI, BmMANN JUmlANIi fNftORATlON IIIT11OBATlOIi

prop08ition of the precedinS precedinl eection, section. If !/ i. It is r.onvenient to use the proposition integrable lb, e), el, then for any ee > 0 we can find etep step intepable on both (6, (4, b) and (b, functions At on (a, b] b) and k. lei and kt lei on (b, Ib, e]c] such luch that functioDl A, A. and lit

with

41(Z) f(x} S S lIt(z) At(x) A.(z) S fez)

for each z~ E [a, b)

k1(x) f(x} S le,(x) k.(z) S fez) kt(z)

c] for each z~ E [b, 0]

L'I.' (ht(s) )tk, I.' I.e (kt(s) k.(s»tk each 1_ than t/2. f/2. (1at(z) - Ill(S) A.(z) )a, (kt(z) - k.(z»a

Define functions (a, e] c] .by lunctioDl /1, /., It /. on [a, 1&.(21) z S b" Al(x) if 4II S :I: /.(21)" /1(:1:) == {{ 1ck.(z) if'" < z S c 1(2:) ifb
/.(z) .. /.(,;) == { kt(z)

't

if II S s S b ifaSzSb if b < z S c. ilb
/l~ /. are step Rep funotions, IunctiOlll, Then 11~

/I(S) ~ /(21) /(:1:) S ~ !t(z) (0, eJ, el, /.(z) S /.(z) for lor all s~ E [tI,

and

(J.(z) - /.(z»a J:r. (Jt(s) /.(s»tk < ;, X(J.{s) (Ja(s) - /.(:.». /,(s»tk < ; .·

Now /. -II (a, e] c] and the propoeition ie is cl-.rly clearly true - /. is a 8tep step function on [tI, functions, 10 for step function., 80 that

Le I.' (Jt(s) (J.(z) -

/.(s». /.(z». -==

J:

(Jt(s) - /.(z»a /.(s». + /.' (J.(z)

t

(Jt(s) - /,(z» /.(s» •. (J.(z)

Therefore . (Ja(s) - /,(z». /.(s»tk < ; + ; I.'Le (j.(z)

- e, f.

This shows that I/ is integrable on [tI, (a, eJ, c). Convenely, if jf I/ it i. intepable on [a, c] e1 ~hen for any e > 0 there are Rep /1, /. OIl (a, step functiOlll functions 11, on [s, (a, eJ c) such ncb that

/1(Z) /.(z) for all s2: E /1(:1:) S fez) /(z) S S I,(s) E [fJ, ['" e) c)

and

(Jt(s) - /.(s». /,(s»tk < eo I.'Le (j.(s) (j.(s) - /.(s». /.(z». ~ 0, X(J.{z) L'I.' (Jt(s) (Ja(s) - /1(Z». /1(S». ~ 0 to

Since

and

(J.(z) - /.(s». /1(Z». - I.' LeI.' (J,(s) L' (J.(z) (Jt(s) - /I(Z». /.(s»tk + X(j.{z) (JJ.s) - /I(S». /.(s».

we have

'4.

n7IfDAIOIlIWAL"...... OJ' CAIAlULU8

111

pmpoeition of the lut Iut. eection, 1 / is intep'able Thus, &pin again by the proposition integrable on both

(0, e). To complete the proof, auppoee / intepable on [G, IG, b), l6, b) and lb, c). 'lb, e). Given e > 0 we can find •a > 0 noh lb, e) c) and [G, c). such that any Riemann

sum [b, c) and [a, [G, cl 0) of 01 width 1_ aum for 1 / corresponding to partitions of [G, b), lb, /(z)dz and I(.)dz than I differ in absolute value from L' /(z)dz, /(s)dz, /(s)dz /(s)dz _/3. Take partitions of IG, cl of respectively by leaa leas than e/3. [G, b) b] and of 01 IlJ, [r., 0) 01 width 1_ than •aand Riemann aurna 8 1, 8, aums SI, S, for / COJTelPOndini correepondinc to theBe tbeH pariltioDL pariltionl. Then SI 01 (0, el c) 8 1 S, 8, is ia a Riemann sum for / corresponding correspondinc to & partition oIIG, leaa than I, and we have of width leas

I."I.'

L'

a

+

L'L"

a,

1lSI81 -- f I(s)dzl /(z)dzl < ;, 1IS,8 • -- J."/(z)dzl /"/(s)dzl < ; ,

IS. + &Bt -

L"/(z)dz\ L'/(s)dz I < : .

Therefore

IL'/(z)dz J."/(z)dz L'/(s)tk + /"/(s)tk

L"/(z)dz L'/(s)tk

I

s~ I /.t!(z)dz 8.1 + I J."/(z)dz L'/(s)dz - SII /"/(s)dz - &1 Btl 8 .+ 8.+ lSI + S, /(s)al
Since e wu any positive poeitive number, we have

Lt !(z)dz + I." I(z)dz - L" /(z)dz. L'/(s)tk /"/(s)a L·/(s)tk. lhtfirdtion. If! If / is ia "...'don.

an integrable real-valued function on OIl the internl in.....

[G, (G, b), we set

r.!(z)dz .. - J.'/(z)dz and, for any c E E [6, [CI, b), bJ,

1" /(z)dz - O.

111 .. G ~l"'~ ~l_~ fm ... """ . ita.R.teA ......, Corollary. 1//it Ott . ." ......, . .weA,...,.., CI, h, b, c ad of 1M ~ /.' J(s)dz, /(s)dz, J(s)ds, /Cs)dz, /.·/(e)ds 1M poinU G, and if -I two o/IM J(s)4s aNt, cNm tAw ma. __ ad anti eNt, ~ 1M third

L'

f.'/(z)dz + L'/(s)a

I(z)dz J.'J."/(s)dz

L'J:

L'

L"/(s)dz. /.' J(s)dz.

The special verify, 10 epeoial cues cuea G - h, b, bh - e, c, and G - C e are all trivial to mr" numbeftl G, lJ, e ...... deterwe may uaume the thne three number8 G. b, e diatinct. diatiDct. The pointe pointa a, 0, .. mine & certain cbed the union 01 _ cloeed interval in R that •.. ....-eel " . e d .. .. tile eloaed subintervala, Ia, b, closed subintervals. namely the interval (min [min (0, b. cl, max (0, IG, b,h, ell &ncI &Del the two 8Ubintervals aubintervala determined by that. which ia that, point &mODI G, s, Ii, h, c "bioh is

amonc

..

ft. . . . . . . . . IR'ISOIU.'ftOIf

I." I(%)dz, fI.", /(:c)d%, .

betweeD between the other two. The exiatence existence of any of L'/(s)d%, L' I(%)dz, /'·/(s)d%, I(s)dz,

L·/(s)d% L ""/(%)dz is equivalent to the existence of the corresponding Le /(s)d%, the proposition teUa f].'/(s)C, I(s)dz, f I(%)dz, tells us immediately that the to

80

exilteDee exiateDce of two of the integrala integrals in qu.tion question implies that of the third. Thus we may .-ume uaume that all the integrala in question exilt exist and it renwllI remains to .e \0 prove the equality, which we may take to be in the equivalent but more qnunetric fonn IIJIDII18tric

L'/(%)dz + /.·/(~dz +

f I(z)dz ... O.

prove this 1ut equality, we note that it does not alter atter ita its sense wider wlder To plOve permutatiooa of a, 0, 6, c which send 0, 6, c into h, b, c, CI 0 nwpectively. reepectively, the cyclic pernlutatioDl CI, h, 0, 6 fl8PeCtively. reepectively. Hence we may . .ume without 1018 1088 oIlenerality of leDeranty or into c, a, _ume 6 is between G 0 and c. Thus we are reduced to the two special CMeI cues that b • < b" < cando> c and CI > II6 > > c. Co The truth of the equality in the 'first firat cue follows directly from the propoeiuon. same uas the firat, proposition, while the second cue case is the 88me til'St, but with. . witb a chanp change of lipa. sips.

A further coDleQuence of the proposition is ia that if a real-valued function /I ia intepable on •a cloHd elClRd interval in R then L'/(z)a CI, b 1'/(%)dz exiate exists for all 0,6 e10eed interval. We remark that if 1/(%) ISM 1S M for all :c % in the tbe clOled closed in tbia claIed 0, " in the interval interval then for any a,

,2

ia trivial if G 0 - II, 6, •a CODlequence CODI8Quence of the fact that -M S 1(%) SM This is I(s) :S for all s% in the interval and the last b, and a conseJut corollary of 12 if 0G < h, last cue and .ymmetry eymmetry of lip sign if a 0 > b. quence 01 the lut

Theorem (Fanda"..",., 7'INorem (Fanda ...."t.I tlaeor.m theoNm of ealeu'..). cab""). 1M IMI/ be aCJ COfttinuoua ccmtinuoua real«dued/ufl,ditm Oft 8ft open ,...,...,ueclfvnt:4iqr& opm iftIen1tJl inIenIGl U m in R and let a0 E U. Let tM/tmditm the/UfldiOft FOIl tlefl.n«l F Oft U lie be " . . by F(s) F(%) - Le/(t)dt/or L·/(t)dt lor aU s% E U. TAm Then F ia if diffmmtiablc differentiable

_'-I·

.11'-1· Since 1 / ia is continuous, F(s) - L-/(Odt L"/(t)dt is defined for all s% E E U. W. We have to ahow Ihow that for any fixed Zt Z'1I E E U

• 1i'(s) F(%) - F(%t). F(:eo). 111m

1m ..... -..

s-So s - Z'1I

-

! () I()

So. %0.

For any z

II

E U, z .. ,. :ee, z., we Mve bave

'(:c) '(z) - '(z.)

~~~:z...

:cz-z. -:ee

I IL"/
-/(ztJ --/(z.)

I

L·/(t)tlC - L" 1(l)dt -/(:r:0) -/(z~ z-z. 1(')tlC _ f~ J~ 1(z.)tlC J~ (J(t) 1(l)dt l(ztJdt /~ V(t) ~ I(z.»dt :c-1Ce :c-1Ce :c-zo z-z. z-z. :r:-:re

I_I

_I J;/;

I.

&iDee I it II conuuoue ICe, liven any Since continuoue at :re, &I1y I• > 0 we can find a a > 0 IUcb lUeh that 1/(:c) -/(z.) E U and l:c -1Ce1 < f.•. Thue ih E U. U, 1% --:reI 2:01 < a I/(z) -/(ztJl1< < •I if :czE I:r: -Ztl and z ,. Zt then for &oy &I1y ein the extremitiee :re and % z"lCe tbe cloeed interval of extremities:lt z we bave 1/(1) I/(e) -/(z.) 1 I < .,I, 10 that Mve tb&t

IL: (J(t) -/(ztJ)dtIS.I:r: -:reI· -/(z.»tlCIS .1. -1Ce1· (J(I)

bave UI8Cl ltatement of the (We Mve ueed the remark immediately preceding the .tatement theorem.) :r: E U, I:r: - z.1 < aand % :r: ,. :re we Mve l:c -1Ce1 bave t.beorem.) Therefore if :c

I

<,

"ICe

IIs ·t~'I':zr: ---ZtILjl_--.•.

,(:e) I'(z.) -/(z.) F(z) - '(ztJ -/(ztJ S

z-a:.

I/-z.

f.

I/-SlI.

TbiI pIOV. the d.red d_red limit ltatement. 1'biI CoreUary J. 111 if 0Q conti,.,"*, CIf'& open ifttBrtHJl Corollary 1. coratintwtU reakalued r«Jl-rH.IlU«l fu'Mlitm jemdifm OR em on interlHll in . 0ca retJl..tHJlued reol-tHJlU«l functifm F em tM ICImB lOme i1&ter1l41 interlHll ",koae ita a, ft, lAm lAM dtere tItere . aiD /tIftclitm I' Oft 1M whou tIeriNeiH if/. tIeriNIiN it I.

II any fixed point in the interval, 1'(%) For if. if 0 it F(:r:) -

fL·/(OtlC l
We recall that tb&t if F' -I then I' F II ie called an tJJ&titltriIGtille oneitlfrWalive or primitive of I. Corollary 1 1&11 that tb&t any continuoue real·valued real-valued function on an open F iI interval hal baa an antiderivative. If H I' ia an antiderivative of I on an open interval, 10 it II ,I' + c0 for any conatant ccmatant c0 E R. Furthermore, any antiderivative of II muet Mve bave the form' fonn I' + 0 : for if 0 it is another antiderivative of I. I, then (0 -1')' - G' - F' - 0, 10 0 - I' F mUBt Dl,uet be conetant, coDltant, by Corollary 1 of tbe mean value theorem. the m-.n

CoroIIar7 ,., fvnditm ~ I' OR CIf'& IaCII Corollary J. 111M 11 tM ....... ~ F em on open iftttlnHll intenH.ll U ita in R 1Itu tM ccmCiftUOUl coratinuoua tIeritHJtWe and 0, b E U, """ tMn lIN tIeriIHJIWe I GfItl Lt/(l)dt F(b) - '(0). F(o). Lt/(t)de - '(b) I(t)dt - F(:c») !t:r: (I: (J: I(e)de

J:L" I(t)dt - F(:r:) iIis CODBtant. Thus Tbue L" L· l(t)tlC 0, lor lOme c E R. In particular oonatant. I(t)dt - F(z) + o. 0- J: L- I{OtlC I(t)dt - F(o) tb&t -F(o). La I(t)dtJ(t)dtF(ca) + that -F(ca). Therefore f Since

F(:r:)) -/(z) -/(:1:) -/(%) - 0, O. 80lUe

0,

F(:r:) I'(z) - F(o). I'(ca). Hence

80 10

L I(t)dt L' l(I)tlC t

0 -

F(b) - 1'(0). F(o). '(b)

1'(%)

118 Il11

VI. RlBMANN mBIlANN INTBORATION INTIIO....TlON

This the computation of integral8. integrals. For Thi8 corollary coro1Iary is i8 a powerful tool for ttie exanlple, Z-+1/(", + 1) has the example, if n is i8 a positive integer then the function :c"+I/(n conti'nuous Z", 80 continuous derivative :c", 80 that

+

L' x·~"thtk -

+

(b,,+11 -- a,,+I)/(n (b·+ o·+I)/(n + t).

Corollary 3 (Change oJ oj tmriab'e variable theorem). Let l.T, ll, V be 1M open ifI.tmHJU intervoll in H, function with continuoua conlinuoua deriooliN, deriVGlive, and R, .,: IP: U -+ - V a differentiable Junction f: V l' ~ function. Then Jor for any a, b E U J: - R a rontinuOU8 rofttinuous Junction. f(v)dv = == J.' L·/(IP(u»IP'(u)du J..<., J(v)dv .• f(.,(u».,'(u)du.. ,,(6) tI(M /.~.)

Let F: V --+ = /;., I(v). /(v)dv for all - R be the fU!1ction fu!,!ction defined by F(y) =y E V. Then F is == I. /. The function G: U --+ R 1/ i8 differentiable and F F'' =defined by O(x) == toI., I(v). !(tJ)dv is the composite G == F 0 IP tp of two differenG(~) .. tiable functions, is itself differentiable. By the chain rule Nle we have function8, hence i8 G'(z) ).,'(z) -/(fJ(z) ).,'(z) for all sz E E U. Hence G(s)G(z)G'(~) == - F'(fJ(~) F'{IP(~»IP'(~) -/(IP(~»IP'(~) /.- /(ffJ(u) 1(IP(u) )ffJ'(u)du )1P'(u)du c, for some 80me constant c E R .. Setting sx - CI fJ we get

I"e.) 1"(·) ~.,

L·

+

f

-= 0, 0,80 G(~) == I.·/(IP(u»IP'(u)du. c= so that G(x) /(ffJ(u»ffJ'(u)du. This last equation holds for all % x == us Corollary 3. ~ E U. Setting ~ = b gives U8

I S. THE LOGARITHMIC WGARITHMIC AND EXPONENTIAL FUNcrIONS. 15. FUNCI'IONS. thi8 section 8ection we develop in a rigorous fashion the familiar properties In this of 80me of the functions which are dealt with in elementary calculus.

le .

. (. dl ~ Definition. If z% E R, % z > 0, then log % x= =: 11 -t-·

Propo8ition. function log: 1% Ix E R : :t~ > 01 -~ R ia i3 differentiable Proposition. The lunction 1/~, it i, is striclly incrMBing, flBBUme3 with ddlog~/th vnth log x/dz .. == l/x, ,trictly 'incre43ing, alSUme3 all valuu valua in R, and 8€ltisjie. rule. ,al-lsjies the rules

+

log X1/ ... z: log % x + log 1/ 11 if~, if x, 11 ~ %

y ...

10K log, - log % - log II log~" - n 101 log s~ log:r;"

>0 if %, II 'IJ > 0 if~, V if s~ > 0, n an on integer. in~g"..

log, together with the equation dlogs/th The differentiability of 10K, dlogz/tk I/~, comes from the fundamental theorem of l/z, or calculus. Since I/~ l/z > 0 if Ie > 0, the derivative of log is alway8 2: always positive, 80 80 log is a .trictly strictly increasing i8 80me some fixed positive number and 1/ 11 = fIX, ax, the chain rule gives function. If a is d Idy t - . a - 1I 1 dy 1 Iog , ,- = -log y =: - -=== -(JZ • a == -s' th ~ dz 11yth dz S'

15. 16.

LOOAJtI'I'IIJIIO AND IIUOJdIrftAL IIXJOIf8II'I'I4L ..,.,enONa LOGAIlITIDIIC JVIIcmo.a

119 18

dloga/d:r ... dlogs/d:r and hence lOla loga -lOIs + e, IIOme that dlola/b -= dlogs/a C, for BOme giVell e - 101 log a. a, 10 110 that 101 G:I CUI - 1018 101 G + 101 •• s. After ce E R. Setting Bettini s - 1 lives changing notation we have chanlinl

so 10

logz1/ logSIl -logs

+ 10111

if z,,,II> > o. ih, O.

In the special cue that. case s - 1/1/ 1/11 this yields 1011/1/ - -10111, -101 II, 10 110 thM zs 1 10 10111 if S." log, log, -logs s," > o. O. 11/ -logs -Iog:l: t 1011/ -)OIS --10111

t

Clearly if z

> 0 then log s' - log 1 == 0 ... 0 • log z 101 zI .. logs' logsz .. logs log Zl == 101 == 1 • lOIS logz' logs' -101:1: -logs logz logs .. - 2101s 2 logs logZ&-log (s'. z) s) -logs' 310gs logzl =-101 (st· -Iogzl logs lOIS - 310ls II:

+

+

etc. 80 10

that logs" - "lOIS nlogs if" if .. - O. 0, 1,2,3, 1,2.3, .... lOIS" •..•

(-ta)) I• •, 10 If "n - 0, 1,2,3, 1,2, 3, ... , then logr" log r" -IOil/z· - log l/s" - -lOis" -101 S" - (-.. lOIS, that

log s· ... - ,,101 n log sz 101

if sz

integer. > 0, "n any intepr.

Binee 2 > 1 we have 1012 > 1011 ft 1012, liven log 1 .. - O. Since 1012" - .. given any Since integera Rt, nt. '" tat such that ",.., E R we can find intepl'l . log 22"t 1°1 " < .., ", < 1012"1 101 2" (simply by taking taking"l < ..,/1012 nt). By the intennediate intermediate value theorem 'Y/log2 < fat). (limply loge - 'Y. ..,. That ii, is, ~ 101 loc there is some ec between 2"1 and 2" such that 100e function takel hu been proved. takes on all valueB. values. Everytbioa Everything desired has

"I

Definition. exp is the inveme inverae function of 101. log, that iI is DeJi'n'tion. exp (s) (z) == - 11 II means sz -101'1. - 101". aen8e since the 101 function is one-one (heiDJ (heiDI I&rictly This makes sense Thie Ikictl1 increuing). notation If' for the moment to avoid eonfull_ in~q). We avoid the notation'" eoDf1Jli0ll with our exiltinl exi8tinl notation for powen. powera.

Propoeldon. /uftdiqn up: R -Is .. .,......,.. cICt.....,....., r""""don. TIN /uftt:titIrt ..... I. E R : s • > 01 it viOl flap (z)/ds it *tdlrI ell "... ....... VIiIA clap (a)/. - exp eX)) (s). l' .. eCricIlrI .....,....... ~, ......... ." llCIluu, lIN ,:uz. NlUf8, IJft4 ncI ItItiIJia ~"..""" (a) • exp up (r> (w) - exp up (s + 1/) iI up (z) i/ s, 1/, E R

8!P (s) _ exp (:I: -,,) exp~a~ exp(a -,)

exp (,) II exp (u) <exp (a»· (~». (ns) - (exp

e "a, , eB q s"ER

q _ . w.,.. iI za E H, R, ftn GR .,..

ISO

VI. IUULUfN IIIIU1ANN INTZOBATION INTllOL\TlON

moment. Then To prove this, forget about differentiability for a mODlent. everything el8e elae follows immediately from the corresponding properties of the log function. (The easiest way to prove the equations is to check each to see if it gives a correct statelnent statement when log is applied to both sides. Every(z» .. thing works out, using the identity log (exp (x» -= z and the corresponding formulu formulas for log.) A. for differentiability, we first fil'8t prove that exp is conZo E R and any t > 0 there exists a tinuous. We must show that for any Ze aI>> 0 BUch (~) 1< .unle such that lexp (x) (z) - exp (Zo) 1< e• if Is Iz -- %01 Zol < a. I. We may . aaaume (Zo). Since exp is strictly increasing, if zx is between that e < exp (So). 101 (.zo) + E) log (exp (xo) (zo) - e) and log (exp (z.) e) then exp (x) (z) will be between exp (Zo) e. Hence we insure that Iexp (z) - exp (z.) I < •e (%0) - e and exp (Zo) Zol < a I by chooeinl choosing whenever Iz I~ - 2:01

+

+

, - min Is. (exp (J:t) (Zo) - e), .),Iog (Zo) (s. -log -101 <exp 101 (exp <exp (Zt)

+.)e) - Zeal. z.1. +

its derivative, Thus exp is continuous. To prove exp differentiable and find ita Zo E R (Zo) -=- Jlo, I/o, exp (z) ... let Zt R be fixed and write exp (2:0) == 1/. tI. Then lim 1/ II - II. and ...... - ... up (z) - exp (2:0) (Zo) ______11<----".11.:...°_ ·• exp 11 - 1/0 ... l'I·1m ----------1I1m - .. z~ - s. ..... 1/. %t ..... log 1/ - log 1/e

_.. .....

1

log I/o 1 Yo lim log 1/ - 10 "'W. Y -I/o "". 11 -1/0

1 1 .. dI -= ZII exp - til - -1- - tI. y." up (z.). (Zo). ogl/(( ) 01" --;,;;;II. ~ 1/. 1/. This ends the proof. The symbol ~. Z" has 80 far been defined only for integral values of n. In case, if ~z > 0, we have logz"::Z logz" = nlogz, 80 that z" exp(nlogz). this cue, x" == exp(nlogx). Hence the following definition is consistent with our existing notation. ~nition. If z, n E R, z IJfdInition. If~,

Propoaition.

z" ... (nlogz). > 0, then x" == exp (n log z).

1/, ft, n, mER, z, For z, x, 1/. x, 1/ > 0, IN tDe luwB Iwwe ~•• ~x,,-t. Z'" Z- == -= z"'" :1:" z· z· - Z"-z-

--ra--

(s")a: z·· (z")..... z"" (zy)1I == s"Y" (ZI/)" .. z"l/"

Loa __ ..L. "-l· .... --,,,* =,.... ...... dz'" -I

~

algebraic identities follow immediately frool from the definition The four allebraic and previoue previous reeulta results of this section. }c"'or For exatnple, example,

1&. Iii.

LOCL\&lTllM10 AJlD AND UION&N'IUL ~N&NTL\L rvNCl'I'IONB ruNOTION8 LOCWIlTIUIlC

lSI

+

Z"' • S(n log~) • exp (m log~) =log ~ m log:.:) Z" z· ... == exp (nlogx) (mlogz) == eXp (n (nlogz mlog:.c) :: exp ({n m) tn) log~) log x) = ~.+-. x"+-. =

«n +

The proof of the last fonnula is an exercise in ill the chain rule:

d d d n exp x) = == exp (n log z)di"(n x) dz (n log~) log x) ... == z· x· • z X nx,,-I. dzZ"' '"" di" up (n log log:.:) = nJ:,,-1.

~.

=:

The rules for fractional exponents are of course contained ill in the last example Zl/' Zlll - ~, ~,since since (ZIIl)1 (x 1/1)1 .. == Zl Xl ... == Z. z. proposition. For exanlple It is convenient to extend the definition of x" slightly by setting O· 0- a; - 0 if n > 0, so that for any fixed positive n the fUllction x· is continuous for % ~ ~ o. O.

Dlffinltlon. •e ... Definition. == exp (1). We immediately recover the standard atandard notation for the exponential function: if z E log e) - exp (~), (2:), since log. log 6 -lD 1. Thus E R then es If' .. - exp (s log.) we may write the formulas of the proposition before last in their more convenient forms forma d ... e- • tJI e" =~, == e....., d.i"1f'''' (ii" et' == .. e",, etc.

approximation of e 111&Y may be obtained by noting that for A rough approxinlation 1 S zs S 2 we have 1/2 S; tklz - 1012 log 2 S S 1. As A. S lIz S I, 80 so that 1/2 S dz/s a matter of fact, we can get the slightly stronger relation 1 "2
hihI

by remarking that it is easy to find a larger step function than the constant leas than or equal to l/x l/~ for 1 S~ zx S 1/2 that is less ~ 2, and a smaller sn1aIler one olle than the constant 1 that is greater than or equal to 1/~ l/z for 1 S~ zx S ~ 2. From Fronl < log 2 < < 1 follows 1 < < 10121 < 2. Hence log 2 < < 1< < log 4, 80 1/2 < log 21 < so 2

<. < e < 4.

It should be renlarked remarked that of course coune we could have obtained all the ...wts of this section differently, Bf4rting results ,taTting with the exponential function. The argument (in outline) is as follows: For any fixed positive integer n the function z· x" is continuous and strictly increasing for zx ~ 0, OJ assuming assuluing arbitrarily large values. Therefore by the intermediate internlediate value theorem theorenl I.\OY ~y poaitive nunlber number has a unique positive n'A positive n'" root. We define rational powel'B powers z > 0 by setting of a number %

z""" x· " ... == (positive nn~a root of z)· x)'" '

common factor other than ::c if m and n are integers with no coounon ± 1 and n > O. o. rulea of exponents, for rational We then prove the various rules ratio'nal exponents. exponeuttJ. If

lSI 131

VI. RlBMANN RlIlIlANN lNT11ORATION tNftORATlON

z > 0 and n E R is not rational, we can find a sequence nlo nl, fit,"', ftt, n., ... of define %. to be the limit of rationalu\lIl1bers rational Ilumbel'll that converges to ni we then define.1:" the sequence .1:"1,.1:"1, %.1, z"', ,1:"', Z"I, ••• , fi1'8t first 8howing showing that thi8 this limit exiata exiBts and is n., .. .... We then verify independent of the choice of the sequence "t, n" "t, fit, fit, all the rules of exponents for arbitrary real exponents. Next we look at the os, for 80me BOrne fixed tJ show by a suitable trick that it i. is differenfunction u.", 4 > 0, 8how show also special choice of bue tiable, and 8how al80 that for a very 8pecial base 0, 4, a choice =~. denoted 8, e, we get the magic formula fomlula tbI'/d.z Qe/dx = ~. Finally we do the log function, which at this thi8 point is easy. Thus we end up with a more nlore natural, but considerably sante results as before. conaiderably longer, derivation of exactly the Bante

,1:

PROBLEMS

1. Compute

=

/.1 = directly from the definition of the intearal, integral, UlllUmilll auuminl only

that this integral Intewal exists.

2. Prove that

J: 1(%)rh ...- 0 if 1(I/n) 1(1/"') C,(s)a

I(s) - 0 for 1 for", for n - 1,2,3, ... and I(z)

all other %. z. an

J:

1(%)rh exist if 1 the function of Problem 1(d), 3. Does C/(%)dz I is tlte l(d), Chap. IV? 4. Let J: [II, h) b)I: [a, .... R and let ec E R. Prove that if

J::,(z J:::

Lt I (%)dz exists then 1(%).

80 10

does

/(% - c)dz e). and these two integrals intearals are equal.

5. Prove that a continuous real-valued function clOMKl interval in R il is Ii. funotion on a cloeed integrable, using only Lemma 1 of 13 and uniform coJltinuity. continuity. clO8ed interval in R and let V be a complete Jlormed Donned vector 6. Let (0, (a, b) h) be a clOIII!d &pace (of. (cf. Probe apace Prob. 22, Chap. III).

(a) Show that the definition of

L' 1I

(%)tk 1 on [a, [II, h) b] (s)a for real-valued functions functiollll

pneralilC8 to functions I: b) ..... V. generalizes /: (a, (0, bl-. (b) Prove the analog of the criterion for integrability of Lemma 1 of 13 V-valued b]. for V -valued funct.ions functions on (0, (a, b). b] -..... V il (c) Using Usinl (b), prove that if I: (a, [11,6) is continuous then

fJ.' I(s)b exists.

I(z)ds exiat.L

(d) Prove that if I: b] ..... V is continuoul then /: (a, (0, b1--+

/.'II/(S) II f/.'/(s)a\\ 1(%)rhII S ~f II!(%) II a. rh.

finite-dimeneional with basis PI, ... .•• , "., II", and if II, •• •,,1" (e) Prove that if V is finite-dimensional buis '" fit ... f.

b], tlten are real-valued functlonl functions aD on (a, (II, 6), then

exIate exists if and only if

r.J:

(f.(s).,. (ft(Z)tll +

r.J.'

J:

f I.(s)a

(/I(S).,. (/1(%)tll

I,,(z)...)dz + ... + /.(z)tI.).

I.(%)dz, •.. exist, in which cue ft(z)tk, ••• , /.' I. (%)tk emt,

(r.

I.(%)v,,)dz .. - (J.' (/.'Ia(z)a )11. + .,. ... + (J.' J.(s)tk) I.(z)a) ... .... ... +1.(z)f1.)• !I(%)rh)tll

Probe 23, Chap. IV.) (For part (e) you will need the result of Prob.

,.,..... us 1 on the interval Prove that if the real-valued function 10ft

r.

Ie, 6) ia ie bounded and la,

I(~)dz . ~ £I(s,. .. increuina (or dllCl'ealling) decreasing) tMa thfJD r. I(z)a emu. exietL Prove that if I: lei, bl- R is increuiD& J: l(s)4:1

il continuou8 except at a finite number 01 of points. points, then is continuous (CI. h) - R is

(fI, 6) ie iatepable OD on Prove that if the real-valued function I/ on OD the interval Is,"J is iDlep'able (eI, b) hI then 10 80 ill is (CI.

IJ:

I r.J: \/(s)\•. II(~) I•.

III, and \ Lt'(Z)a III. I (s). \ s

(a, 6) is ie iDtepabie iatepable OIl Prove that if the real-valued function I/ on the interval (a,"1 (a, /'. Using the identity (f fI 2/, prove that the lei, b) h) then 80 is 1'. (J ,)t ,)1 --I' ~, product of ~wo integrable function8 functionl is il integrable.

+ +,., + t,

+

continuou8 real-valued fUDction function on the interval la, (a, 6) b) IUeh Prove that if If is .. a continuous aueh z E (a, (eI, 6) hI and /(z) I(z) > 0 for iome -II E l., la, &), bl. then that I(z) ~ 0 for all % tMa

J: I

,(z). (s). > o. O.

continuoul real-vtJued real-valued function on the InterYallo, Interval Ie, &J b] tMa then Show that if 1 I il ..a eontinuoul

J: II

(z)a --/(E)(h a) for lOme (s). I (E)(b -- CI)

[G, b) hI (mean value theorem for EE (0.

intelate-

grals). Show that if 1 I il is a continuoUi ,function continuous real-valued real-valuedfuncUon [a,6), ~ 0 for all z E (a, 6), then

fI(z) (z)

~~ ~i.!!

OD OIl

the iDternJ iDtervaI 10, la, 61 bl aDd and

(J: (r. (J(S»ttdz)1I1t (j(z».d.1:)a'" - max lJ(s) : sz E la, (G, 6)). bU. I/(z) :

e

Show that if 1 ie a continuous real-valued function OIl ADd J is OD , ,-. E R : •II ~ 01 aDd lim/(z) - c (cl. (el. Prob. 8, Chap. IV), then

...... ...-

lim!! (·/(t)dt fa I(Qdt -_ e• c. lim ...... z). ... _zJ.

Let (a, (eI, b) and (e, continuous realIe, dJ be closed intervals in R and let I/ be •.. coatiDuoua r) E E' gt : z E (0, (a, 6), b), r e E (C, dJl. valued function on I(z, II) c111. Show that the func-

tion ,: (e, II) --+ R defined by,M tion,: Ie, dJ by,(y) -

r.J:

'uac-

I(s, 41 iaII 00IdiD-. I(z, r). r)a for all r E E Ie. (c,1II continuous.

Prove that the real-valued function on

J: I(~)da: I ie is uniformly contiouOUL (z).

cao, liD caa, hD

which .-cia .ada aD1 aD)' I iato

uniformly continuous.

.w.n

Prove that if H l' u and 11" are real-valued fUDctioDl OD an aD opeD ,·01 functioaa OIl open of •R containi.. the interval (a, la, b) h) and have eoatiiluoua continuous cIerlvatl derlvatl.... theD &hell containiDl aDd if l' u and , hay. (intfcration " (intearation by parte) parts)

L'II(.>U·(.""

L'

/.' -(.:).'(.:). 1'(Z)II'(II). - -(6)11(') 1'(6)11(.) - u<.)lI(a) l&(a)lI(o) - /"I(a)v'(a,.

Prove that Provetbat

f· ,.m:;a

C'J:

tit tlu J.(./v1"+iI "1 v'1 __ ,.. p. +... le - .•,I1 + at tit

flu

aU •z E •. for all a.

8hcnr that if U is lID aD open interval in R and the ...... R baa a conCOD19. Show tbe function /: I: U ..... tIauoua (" (a + 1)" derivative on U, then for any a, a. IIb E U we bave tinuoua have I(b) _ !(a) 1(6) J(a) +/'(a)~I+1'(a)~I- a) +/"(a)~I+1"(a)~I- a)t a)' + ... + f(·'(a)~I.r"'(a)~I- a)·

eE

(6 - %). (b s)- /,,*II(s)a pa+'I(S)tk

. + "I 20. cootinuoua fUDCtion function I froID from an iDtervaI interval la. la, 61 bl • .A .A eww CW'IJ8 '" ' " • tIMIric tneri: I'fItIa IJHIC8 B iI a OODtiDlIOWI iD R • into iDto B; ita ",.",. lattA iI in is Lu.b.

{tcl(J~I)./(~) : ~ Sa, Sl, ... .. , s. {t~(J("",),/(s,) Sjf iI a partition of la, blJ}. }, 0'

..(%», where if tbia Lu.b. . . . Prove that if B - .. B- aDd 1(%) /(s) - (J.(%). (JI(S), •.. ,J ,I.(s», la, •• •,1.: la, (., 6) 6J ...... R are coatinuoua 00Dtin1lOWl functiot18 functiOhl which have bave oontinuOUl 6oDtiououa derivadaivaIl• •••,/.: tine on (•• (a, 6) that exteDd to continuous OODtioUOua functiona functioaa on la, bl. 61. then the curve baa ti.,. lqtb aod and tbia tltillenith 1eaIth IeDIth II is 000

r.J:...t"

(A'(s»t+ + (J.'(%»' (/.'(s»ttk. C/a'(s»' + ... ... + a.

11. II. Comput. Compute
~I.+.~;" +",whereA:eR,II>O ~1'+"~;" +".wberel:ER,l:>O

(b)

I . 1 l\ ( _+1 !!: +_+1 + ... +~. + W· ~("~1 +"~I+'"

Do Show that thM for .._ - 1,2,3, 1, 2, a, •.• , the tile IlWDber 22. 8bow number 1I 1I 1 0

•••

_n.. _

1+1+.+'" +;-IDI. +;-Ioc .. 1+I+i+'" iNn.... MIl beDee __ that the ....... that tbe IICpleIltll of of . . . DWDbera coo.,... ~ 0 aod COD8Wat). .......... OOD"- to a Iimlt ..".. aDd 1 (Euler'1 (Euler's eoaataDt). 13. IllCh that I' --I 28. Prove that the only flIIICtion fuDction I: R .... - R aaeh I &lid uad 1(0) - 1 is liVeD

II politi pGIltive, .... it II tIIat it

br 1(S) -t/'. -,. by/~) K. II. Provethat ProvethM

+.) > -I, -1, with equality if and only if s - 0 (a) . Ioc. (1 + s) S s for all s > tI" ~ 1 + +.s for all s, with equality if and (b) tI' &lid only if s% -- 0 (c) lim _1 (0) z) _ _ toe (lz + .)

_

_.,

_)"a --lim +!\. + As)"A lim _ (1 +!\"J (e) ..(r" toe z% ifif.s > o. <e> lim .( .... -- 1) - 10K O. -

(d) lim .... (1

_.. -

%

tI" tI'

> • the fuotion i••Itrictly triotly deoreuinl decreuiog and that it pta pte functioll ~ i. arbInrily . . to aero, beDoe arbIlrarIlJ ..... heDoe that

•• II. Show that for lor •

!=. S!_O ~s_O

(a)'" (a)

11_>0 i,.>O

toe s - 0 if ex_ > 0 (b) lim .. toe.

!:...sa -

.!: (e) lim _tI" 0

aD)' .. • E R. for any

......... us . . Define DefiDe I: R ..... - R bJ by ••

.... ,.. ,-I'" {o

/(z) - { I(z)

ifz>O if s > 0

ifsSO. ifs:S;O. Prove that II has derivatives of all orden, with Ita) baa derivatiVt!l I"" (0) - 0 for all ft.

27. U ca, 6 E R,. < 6, and I: Iz Is E R : a •
J....

,... •

Show that if , it another eontiououa eontiDuoua real-valued IUDCtion function OIl on the . . II &DOther Is E R : /I I/(s) I S aU s21 ill in this eat set and aod 8 < Sz ~ S 6111UCh 6111lCh that 1/(21) :s; ,(z) for all

!I:..

I:..

!J:..

,(s)cla: ,(21).

aiata, then f..,/(Z)tlz I(z)cla: exiBta. show that I(z)cla: uiate exiate if there exiate. Henee Hence IIhow 1(21). aiata an aD II CI < 1 I such 8Ilch that (z - .)., exists o)-J is bouDded bounded on (a, (fI, 6). 28. If z ~ .1at ...... R is a contiououa continuoua fUJICtion, fWlCtioD, define the II •a E R and /: I: Is Iz E R : II:

'''''''al

.

......

_,...,.,......, J..... /.-/(21). lim if this limit exist. _prop;, tim t/(z)., J.·/(Z)U, exists (el. (cl. Prob. Probe .. . 1(z). to be.....+. 8, Chap. IV). Show tbatif eoDtinuoUi real-valued -valued function OIl on that if , is another CIODtioUOua

_I, if 1/(21) ,(21) for all s21 ~ ., Is E R : 21 z ~ /II, I/(s)lI :s; ~ ,(z) /I, and if

L. .

,(z)a ,(z)cla: exiata, then tIleo

lIhow that L'" 1(21). exiatll if there exiata exiate ff........1(21)= uiIt&. Heaee Heace Ibow /(s)cla: emu.

IIlCh that. that rf z-f (s) (21) is ia bounded. aueh

L"'/(Z)cla: exiata

an CI II aD

>1

CHAPTER VII

Interchange of Limit Operations

proeellllel we have The various kinds of limiting Proeellel .Btudied (limit of a sequence eequence of point. pointe in •a metric ..... .Btudied 1paCe. integration) do DOt not. limit of a function, differentiation, intepation) atwa,. always occur ocour singly. aingly. In a& given problem we may be ealled upon to take one kind of limit, then another ano~ kind. called auch probleme problema the order in which whioh the operatiODl operations are In such . performed ia is naturally of importance. We bave have already applyins the two operat.ioDl operat.iona treated such a problem in applyiftl lim and Jim /1, /a, ... ..• lim to a 8eQuence sequence of functions lunetioDl II, lit 1.,/.,

"... "... from one metric space into another, ". heiDI beiDl • pain' "...~ ~,.

from one metric apace into another, ,. a point of tbe the firat first metrio apace. If II lim I.(P) exiat.a for each

..,. ......

.

exn

..... ......

n - 1,2,3, ... we let get a sequence Um/.(p), eequenoe 01 pointe Hm/.(p),

fa -

lim /,(p), IeCODd metric &Del /,(P), lim /.(P), •.• • .• in the eecond metrio 1paCe, apace, and

...."

.,...".

. .PI

. .PI

-....

we may be able to take lim pointl. Jim of 01 this sequence of pointe.

....-

exid~ On the other hand the limi~ limit function lunction 8m/. lim/. may exiIt-

-..... .erea.

and if it does we may be able to apply lim to the limit Iimi$

function, &pin again pttilll getting •a point ill in the IIOODCI I8COIld mekie functioD, JMkie space. However it may happen tbM QJieNrthat all aU of 01 ~ open.. tioDi ADd we arri...·., tiona can be perfonned performed and arrive' M dilerent &DIWel'I in the two ~. CMeI. In OIle edNmely im.portaDt import;ut &Dlwen one exveaneI1 cue II caae thie this cannot happen, 'or lor we have PIOyed pIOved thM_ if /1,/,,/., is a uuiform17 uniformly COIlV8J'l8llt eequence8of~ 11, I., ... is. COft~ fl8tlU8DC' of-.... tinuoUl tim tinuoua IUDCtiOIUl,then functiona,then Um I. is aIIo OODtt_, COIltiDUOUl, 10 that ,

't,

...,.. «(lim/.)(p» ....

,

-....... 11_

(lim/.){JIt) lim «(lim/.}(p» - (Iim/.)(pe)

"PI

-

....-

--

..........

(Jim/.(p». --1im/.{JIt) Hmf.
.

III

ftI. llI'IWIlCIUJfGII or LIIII'r OJPaATlON8

In this chapter we prove a number of similar results for other pairs pail'S of limiting processes. No attempt will be made to be systematic; we only intend to provide lOme BOrne especially useful results. At the same time we take the discU88 the meanings of theee these results for opportunity to discuu series, which is ill how infinite sequence. sequences usually infinite seri., ariae in practice, developing the theory of infinite seri. series arise sufficiently for the purposes of calculus. An exposition trilOnometric functioD8 functions is liven given .. an "1 eaay of the triaonometric application.

11. INTEGRATION AND DIFFERENTIATION OF SEQUENCES OF FUNCfIONS.

u .. it a aequence Wemann integrable real-valued funcU 1.,/" I., I., I., .••. leQuence of Riemann t.ioDI ticma on a .. cloIed cIoeed interval [ca, (0, b) in ft R and I., II, II, • •. converges to the on (0, [ca, b), can we assert function IIon usert that

'I, ,., ,., ...

f.,.(z)th? J:f./(z)th I(s)tk - ~ J: I.(s)tk? The followinc followilll example shows that in general we cannot, not even if II, I., I., I., •.. are all continuous.

1, 2, 3, ••• . .. let !.: ,.: (0, 1) 1] - ft R be defined by Fig~ For" For ft - 1,2,3, ao CI. (f. can be defined analytically by setting setting/ .. (z) - 4nll sz for 0 ~ S zx s ure 80 !.(s) ~ 1/2n, !.(s) I.(s) - 4ft - 4ft's for 112ft 1/2n < zZ S l/n, !.(x) , .. (%) - 0 for 11ft l/n < % S 1). 1/2a, ~ lin, x~

J.'•

I.(z)dz - 1, 80 BO that lim For euh IICh ",I. ft, I. it continuous and /.' !.(s)ds t ,,-.cD

- J.'.

I.(z)dz 1./.1 !.(s)ds 1

.-.co ••

1.

On the other hand, band,l-lim/. I - lim!. -- O. (For c1ear1y!(O) clearly 1(0) .. -= 0 and if z% ,4 J4 0 then

....I.'

!.(s) n > lis.) !(x)ds .. lim' 0 !.(x)ds. I.(z) - 0 if if" l/z.) Hence f./(z)dz - 0 ,4 '" .-.cD lim/"/ .. (z)dz. • If, however, 11,/.,/1, Is,!.,!.,

...

converge. converg_ uniformly, there is no trouble:

r,.....m. Let., I..c 0, "b E ft, R, GQ <", < b, tmd and 1st II, !I, I., !., ,., ... Q uniformly TIMoN,". l« !., •.. be G ....,.,.,.,., teqUmce rtGl-ualued !unctiona lunctiona on [G, (a, b). hI. Then .......,.,., ~ 01 o! continuoua r«U-t1calued

---

.....L'

f' (Iim!.(x»ds -lim !.(z)ds. f. }. (~'.(z»th - ~~ L·,·(z)th.

11. '1.

O . . . .UOII.OII OPIl&A'I'IOII8 OIl UQIWICIIII DQVIlNClII8

119 lS9

111

s; s.

l'Iov . . 30. Graph of the func:t.i.oD fllDCt.i.on I. of the example on p. 138. hou.

-

Let /I - lim/•. lim I•. Since Bince each /. I. is continuous oontinuoWl and the

ClOnveqeDC8 OODV81'pDG8

it

uniform, I/ is continuous. integrable on (G, [a, b). b]. By tbede8Q1 ihede6ni.... oontinuoWl. In particular I/ is inte&rable tion of uniform convergence, for any Ee > 0 there exists a positive intepr poeitive in ...N such I!{:c) -I.(z) - /.(:c) I < e/(b E/{b - G) a) for aD all :c E (4, ill. auch that if "A > N then I/(z) ~. e b]. We then have the inequalities inequalitl81

•E _ ~ /(:c) -!.{:c) ~ _ E_ e __ --,,-S/(~) -I.(~)

S-,,b-a

-a b-a

-G

aD :e~ E (4, for all [a, b), b], which imply -E S ~ -e

- /.{:e»ds S e f:J: (j(:e) (J(~) -I.(~)}ds

E

or

If /{z)ds - f !.{:c)ds IS This laat last inequality holds for all" all A

.... f.

E.

> N, and therefore

f:.

lim ~~ L·I.(~)ds !.{:c)ds -- L·/(~)ds. /(z)ds. REMARK. The same auume !. RIIMAax. earne theorem holds if we do not ... ume that each I. oontinuoWl, but merely Riemann integrable on (a, me is continuous, [a, b). Indeed the .. same This call call be proof will hold once it is shown that! that I - lim!. lim I. is integrable. Tbis

.-.....

done easily using '3 of the laat uailll the criterion of the proposition of 13 last chapter,

140

m. m.

INTIIRCRANOB 01' OP LIMIT OPliBATlON8 OnBATIONS INTJDItCRANOB

as follows. Given any Ee > 0, by uniform convergence we can find an intecer integer n such Buch that 1/(2:) f.(x) I < e/3(b - 0) I/(x) - I.(x) a) for all x E (0, [a, b), 80 that

~ a) S /(%) (%) + 3(b ~ a) I(x) S /.. I ..(x)

/I ..(%) ..(x) - 3(b

for all x E E (a, b). Since f. I. is integrable on (0, (a, b) there exist step functioDi functions luch that (/l(X) .. (x) S "(2:) ,,(x) for all z E [a, [a,") '1,,.,,, on (a, b) such f'(S) S I/.(z) b) and ./3(b - a) and ,,+ ". + ./3(" - a) are ffc,,(I:) (,.(z) - ,.(s»d:e 'I(X»d:e < ./3. Then ,.'1 -- ./3(" %

~

ltep functions on [0, b) luch [a, "1 IUch that

e 'I(X) a) S I(z) '1(~) - 3(b ~ _ 0) ~ f(~)

e

,.(x) + 3(b 3(& ~ S '1(2:) - a)

for all sx E (a, [a, b) h) and

«,.(x) + 3(h ~ a»a» J:r. «,,(%)

-- (f1l(%) ('I(Z) -

3(b

3(& ~ a) 3("

a»)d:e »d:e < ..•.

By the proposition quoted, /J is il integrable on (a, b). "). To prove an analogous result for the differentiation of the limit of a eequence of differentiable functions ODe has to make slightly stronger sequence &88umptions. assumptions.

Jr, /., It, fa, la, ... be a Bequmce sequence 0/ 01 real-valued real-valtwl lu'ftdioM Theorem. Let fl' fundioM on an open continuous derivative. 8u'IfPOB' Suppose tMt 1M aequence interval U in R, each 1uwing having atJ continuoua tAat "" N, It', . . . conver,es 801M II a E U 1M aequence /1', /.', fa', ... converges uniformly on U and thllt thal lor lOme 11(a),I.(a), '.(a),, ... converge,. converges. Then lim!,. lim/.. Giata, existB, "iI diJIermtiabk, diJferentiabk, and /1(0),/.(0), '.(a) aratl

-...

".

...... ---

....

(Iim/,.)' lim/,.'. (lim/.. )' - lim/.'. l1li

.-.

By the fundamental theorem of calculus we have

I." 1...'(t)tIt K/. ' (t)dl .. , .. (x) (%) - /I....(a)

(a)

oc / ..

-

for any ~ -= 1,2,3, .... Let Iimf,.' g. By the previous x E U and any 1) n'" lim!.' .. - (I.

...-

.

.~

r.J:. fez)

theorem lim (j.(z) (f.(z) -/.(a»· - /.(a»· exists for any z E U and equall equala ~

,(l)tM•• (I(OtU

Since lim!.(o) Iimf.(2:). Setting lim/ lim!.(z) lim/.. (a) exilt8, exists, 80 does doee lim/.(x). .. (x) .. - f(z) we have

.... ......,

It.... "4ClD

I(x) - /(a) I(a) oc .. /(%)

.... ,.. . .

,(t)tIt f:f ,(t)dt

BeCOnd UBe use of the fundamental theorem of calculus gives giVeR for each zx E U. A IeOOnd

I' .. w be proved. - I, which is what what. wu was to

12. 12.

IN"""''''' III...... _

I J. INFINITE

1ft 1ft

sa...

it\11".

ai, lit, If aI, tit, lit, CIt, • •• i8 is a sequence of real numben, numbers, by the ~iIe ..... -w.

al+IIt+Ge+ al+ tit + CIt + "', ''', also denoted

+

+ CIt, Ge, •••• The tenna 01 the latter we mean the sequence Os, 41, Cli 41 + tit, Cli 4t + + CIt tit + CIt, terma of pcartiGl "'tria 01 •. If A E sequence are called the JHJr'tiallVtM of the eeri eeriee. e R, we lAY 8&y that the c:onvcrgu to A if the sequence 01 infinite eeri. eeriee CONIergU of partialBUme partialwme conVe1J81 COD",,- to A, that il ill if

--

lim

(CII (41

+ CIt tit + ... + ca.) (1,,) -

A.

If the leri. series convergee customary to call A the IVm oj 1M .... ...... converges to A it is iB CUltomary ...'" oJ (although thill thil is iB not a lum (althoup Ilum at all, but a limit of BUmB) lume) and it iI is CUItoInary cudomary to write at + tit + CIt +

... -

A

or

+

(This 4t + (Thie somewhat awkward convention, whereby we use \lie the aymbol Os + a. wm, if the latter exilte, exiIta, CIt + ... to denote both the seriee eeri. and ita BUm, cauaee confusion, confueion, lince i8 usually ueually clear from the context whether rarely caueee Bince it is aeries or itallum ita lum i. the series is meant.) If a leri. seriee converpll converp8 to some _real number, be~; contrary ._ . . the is uid said to CONIerge, the eeries aeries iB converge, or to be ~i in the contnry series is divergmt. aeries divergu, diHrgu, or i. diver,ent. Similarly, if 11,1.,1., ft, f., f., ... is functionl on a il a sequence of real-valued functiona lpace E, by the infinite series leri. metric space

tit CIt

11+1.+1.+ fl+f.+f.+ ... of functions fl'/I + lit f., II fl + I, f. + I., ..... 8&1' 01 functioDl 11,/1 . .. We .Y

we mean the sequence that the eeri. ~ at cat 1', for a certain ,peE, E B, if the __ eeriee COfUIergft aeries /I~) It(,) + Ih) + 1.<7') fa(p) + ... CODVerges; converpllj otherwise otherwiBe the eeri. eeriee II I. + I. + I. + ... iI eaid to diHr"e diNT,. at cat p. The series aeri. iB uid to "",..,." said is said ~ Oft em B (or, more mon limply, to "",..,.,,) if it converpe conve. . . at each point 01 iI •a _. to~) of Bj B; in thiB this cue tbe1e there it real· valued function /Ion on B E BUob weh that 1(P) /(1') iI ie the awn lIUm of the eerieI 11(,) + - 1,(:,) f.(p) + 1.(1') /,(1') + ... for each l' E E, COUl'Ie write 1.(1') S, and we of COUl'IIe

+ + + ...

+

+ ...

e

I.+/.+f.+·" 11+1.+1.+'" -I. Finally, we say uy that II ~ OIl FinaUy, fl + +/.I. + I. + ...... CIII&I1er,_ convert. tmi/~ on B B if the sequence f., ft + 1.,/1 f., I. +/. f., . .. convergel eequence of partial sums lOmB 11,/1 + /. + I., converpe uniformly E. on onE.

'flI. INTJmCIIANO. O~ LUOT OPmlA.'ftON8

141

Ex&MPLIl. Eft, ExAMPLII. If aa E R, Ilal al < 1, then the "geometric" series

+ + G' + ... converges. In fact, since (1 · ·· + a-') 0-- -== 1 1 -(1 -- 0)(1 G)(1 + GG+ aal + ...

1+G

a ~ GA

..

0,1

l

1 )

0" a"

we have

I-aI-a" 1+0+0 +11....1 .. I +G+o11 + ... +a-· -I-a I-G 10

that.

1-01 •1 - aA ~(j" 'r G- -lim------lim - - - - - - . :::a "... 1-0· ~ -- I-a I-G

-

.....

(We bave 0," have heN beN UIed ued the fact that lim G-

,3

-

-

'0,1 < 1. Thia This was wu proved lal

0 if

t.he end of 13 of Chapter III. Another proof is obtained by noting that. at the that lollcal 1011 ai- -- "ft 101 Ial is neptive negative and pta arbitrarily large in 1001alill abIOlute ablolute value U u tI inc.....) iDCl'elllll.) Lettilll s denote the identity function on R, u usual, we pt the analoaoua functioDl on the metric apace analoaoue atatement statement for the aeries of functiolUl (-1,1): . I1 l+z+zI+~+ .. ...· - - - . l+s+sI+sI+ I-s 1-~

The elementary facts about infinite sequences of real numbers numben can be tranalated immediately into facts fac .... about infinite series of real numbers. For translated

.em.

lince an infinite sequence can have at IllOBt example, since most one limit, an infinite aeries can have at IllOBt Bum. Aa serieB most one 8um. As another example, &ince since a sequence of numbers eonverpe converpa if and only if it ia real numben is a Cauchy sequence, a aeri. series of converpa if and only if iita .... sequence real numbers converpe eequence of partialauma partiallums ia is a Cauchy IIIqUence. Since for "ft > til '" the difference between the "," aequence. m" and ft" partial eerieI •• + lie + · .. iais ..... aurna of the serieI aulll8 41 + CIa CIt + tJw.+l + ..... «'+1 + ... ·.. + .. 0..,, we have the followinl ..... t. result.

+ . ..

Propoall.". TAe aeria wiu 0/ number. a. Propodtlon. 0/ real number' 41 + CIa Gt + .. til + ... corwergu converga if tJfttl , if, GnY •e > 0, tMre i8 such that if 71n > '" m~ ,., _ Oftl, ii, giNn. pm -II it a(J poaitive poaitir1e integer N auch N IAeft N """

1..... + ..... + ...

+ .. 1< eo

followilll two corollaries corollari. are immediate. The following

"Iim-..... ..

1/ t.U aeriea wiu 0/ real number. 11 1M number, nma.-o. -O.

CaNIIary I. CoroUary

~

G. Cll

+ .. Gt + .. 4a + ... ··· COftIIerge., converge"

..

11. ...._ _

.

ItS

Il.am. .- -

161

• • CoroIIory J. 1/ ~ ca,. . " ~ 6. are .,.. in/lftiU ifllI:law fIria atria of"'" 0/ ,., ...,.•. "umbtr. ,avd& e-ou.r" awA IIttIl a. ca,. - 6. vIaeraaw "n U ~11 ~" lar,,) lorge, lAm mae """ atria eouerf'l etmII6f'fIU 10 IAal b. ...... tAm i/ if one to

11'" ." 116.

doa tIae oU&er. dGatMotMr.

..

•

En....... _ _ ~ 0EuIO'LJD 1. The pometric geometric eeriee CI· doee does not converp converge if 101 1111 ~ 1 by the 8nt fim corollary.

1.. ... - 1 +1.. 2 + 3 + ···

Ex.uIPLJII ExAMPLB 2. The "harmonic" aeries :=I f'1..ft :;sn diverges. For whenever" whenever n - 2t.,. we have diverpl.

+.

2".

1 1 1 1 1 1

a...-I+ta.+t+ _+1 + c&.+1+G.+I+ ... .. • +a.+ca,.- ",+1 +

m+2 + ",+2 + ... +fi"

~1..+1..+ ~1..+1..+ f t" " "

... +1.._,,;.1...1.. +1...,,;.1..-1..

,ft , 22 ''

"ft

propoiition. contrary to the condition of the propoeition.

(1) (I)

. ",;" E•

..

..

_1 .-1

. .. .-1

• • 1/ E E a. _ E E b. corwer(lmt ",iu oj real ca,. cmd 6. are on CMWer,ent aeria 0/ ,etJl number., tAm lAm "" U&.

Nriu

.-1 -I

(ca,. (a.

6.) ia u alIo e&lao couergeAt + 6.) ~ -cmd

...E
.

(I) 1/ 11

It .t... i. IIa

ia

0II

(ca,. + 6.) -

..

E + ... E _ ....

ca,.

b••

couergeAt aeria atria 0/ 0/ rtGl retJl "umber. R, ~ "umber. -tmtI c E ft,

lAm tAM

.

~ CCI,. ia ~ and

.

....

..

....

Eca.-cEa.·

The proposition is imnlediate immediate from the third proposition propoiition of Chapter III (pap (page 48). 48) .

..• """ E a.

Propoell"". 1/ Gl, CIt, Ga, • • • are f&ORnegative real

atria

tJrbitrGriz" or it .w Aa arbitnJriz,

.... ca,. ~ or • -1

c:tnaVerfIU

"umber"

13

of

tAm eiIAer 1M

large" partial",,,,,• panitJl Jar' IUfIII•

For the sequence eequence of partial 8un18 luml is increasing, hence conveflent convergent in cue caae it is il bounded from above.

144 1"

VU. OJ' LIMIT VB. JNTEBCBANOZ JNTZlICIIANO& 01' LDlJT OPIJRATlON8 OPIIBATJOIfll

Proposition PropotJition (Compariaon (Compar"on test). tat).

1/ If

t a.. and t

__I ..-1

a,.

..-1 ..-1

b.. lire are infinite uri,. ,eri" b.

of real number' 8'UCh that I111a.... 1 I ~ b.. == 1,2,3, 1, 2, 3, ... and ond number88uch b.. for n =

t

conver,_, b. converga,

..-I .-1

, For let E be any real number greater than zero. By the fint first proposition of this section there is a positive integer N such that if n > > m ~ N then

+

+ ... + b,,1 b.1 < E.

Ib",+1 + b ..+1 + Ib"'+1 b"'+1

E.

Thus if n > m ~ N then Thusifn>m~Nthen

+ ... a.. I s ~ 111".+11 .. · + 0,.1 10..+11 + 1110.+11 1"-+11 + ... + 1a..1 10..1 ~ b00+ b. < Sb.. +1+b-+ +b.
1~1 + 0"'+1 ~I + 10.+1

Il

aD

,,-I .-1

Corollary.

Under conditiona of the propoaititm, propolition, Uruhr tM the condititma

tllal~tb.. tb". IIta..IS a-I ,,-1

,,-I

••1

+ + ··· ... + a.. al s~ a,. is such that 1111 Ial + lit o. + ... ·· · + l0..1 + ....·· + 1IIIaI ~ Eb., Ilall at I + I11111 at I + a,. 1 ~ b1 + bt b + ... + b. btl S ba, and since the closed ,,-I interval [ - Eb", b., ..-I••E1 b,,] b.] contains each partial sum III01 + litfit + ... ··· + a.. it --I ,,-1 111 + Ot lit Each partial sum al l

t

_1

t t

also contains the limit al80

l

t

CIa of these partial sums. a. BUms•

a-I ..-1

Definition.

If ai, II., at, /It, aa, lIa, ... ••• are real numbers, the series

-

• I: IIa E fit. is aid said to a-I

... IIIaI

t•

absolutely convergent, corwer,mt, or converge COfWIr'ge absolutely, be ab.olutelll ab,olutelf/, if the eeriea 88riea ~ convergent.

...

10.1

ia is

a-I

According to the proposition before last, a series of real numbers

1: sums of the Ea..IIa is absolutely convergent ifjf and only if the set of partial partialsumB

.-1

,,-1

1: E CD

a.. I is bounded. By the comparison series __I Ia" cOlnparison test an absolutely convergent

"-I

(',omparison test is actually a test for absolute series is convergent. The eompar1son convergence, and the following "ratio test" tat" is essentially a special cue case of it.

12. Prop08ition (Ratio te.e). teat). Prop03itioft,

11 1/

urn..........

UfnlQft....

I. 145

tto." a. u an on infinite inJi,ni/.e aria feria 01 rumuro real t'&DfLUrO

_-I a-I

number. and if tAat Il..... aI.. l:S fill il there exis", ezi8t. Ga number p < 1 lUCIa auc1a that ,,-+1/a.1 :S pP lor GIl auJliciently l....aI.. 1~l/tW nU/icientl1l large ft, n. tJum, then the aeri" Beriu converga ~ abIolutelJl. 1/ 111,,-+1/a.1 ~l/or aU au,f/iciBntl1J 814f/icientl1l large A. "" the .,ia BerNs diver,.. diverges. t.Jll IM,e ft,

ab8oluIel,.

phraaea "for allsufBoientJy alleufticiently larp ." A" For the proof we may replace the phruea by "for ufof all n", since lopping off the fint aU A". first few termI tenna of the 181i. aeri. does not 1,,-+1/a.1 S S p < 1 we have affect its convergence. Then if ItJ.+Jo.l

10,,+11 S pIa.l S pi Ia.-II s S ... .•• SS p"IGII, palall, 10,.+11 ~ pia.l ~ p11a,.-11

E t a..a. follows from c»

80

the absolute convergence of

+

a-I

compariaon comparieon with the

+ ....

seri. 1Iall pi IGsl all + .... On the other hand, from the geometric series Gil + p IIall Gil + pli atatement 1a..+1/0..1 10,,+1/0,,1 ~ 1 comes the fact 8tatement

1a..+11 /0,,+11 ~ 1a.1 ~ Ia.-II a.-I I ~ ...

~ lalt, lall.

the sequence ai, Oa, ••• doea not have aero proviDl that Gl, CIt, Ga. lero .. ita limit, Provilll aeries diverges. the series

80 10

•

Corollary. 1/ E a.. lUCIa that tAae lim I.... J.. I 11 1M the aeri8, Berie. 01 real number, number. I: a. i, i. auc1a ',,-+I/a.1

_-1

..-1

.,.i"

a4CD ....

0010'' "'11

exi.t. aM and iI ia lu, luI than (greater than) one tJum, then the Beriu converga COJ&VeTgeI abaolut.lll ui,u (diverge,) (diverge.) • ia leaa For if the limit is tell than 1 we may take the p of the proposition propolitiOD to I, while if the limit greater than be any number between this limit and 1, Iimi~ is If8&ter one the proposition is directly applicable. The series

111 1-~+~-~+ 1 - - + - - - + ... 2 34 4 23 :

ia an example of a series of real numbers that is conve.... convergentt without beblc is bemI abloJutely _ it Iinoe ~ absolutely eonveflent. convergent. The . aeri. is DOt not abeolutely abaolutely CODV-aent converpnt Iince the

I: l/n, E-1/., .... which is II known to be divergent; the convergence of the _ _ ......., it • nault of •

the harmonio ..... _ _, ~ harmonia .-a which known to be divergent; the convergence of the .... atatement on "a1tematin1" "alternating" ..... aeriee. the following more pneral statement

corresponding 181i.. series of ablolute abeolute valuea values is correapondinl

deer""

Propoa'tion. ai, fit, Oa, ••• be G a decr~ ~ 01 0/ poNiN ~ •AumlHn ProIHM'tion. Let Le' (11, tit, Ga, .,.,.,., Then IJae the Nrfa Beria converging to zero. TAm

t•

CIa E ((-1)-1 -1)--' a. a-I .-a

lID

Cll -(II

fit tit

+ OaCIa -

converga ~ to IOfII6 some poaitift poritive number lea, luI tIum than Gl. Cll.

a. ..

+ ... ···

1~

....

III"J'IIJICIIAIfGII 01' LlIlO'l' 0 . . . .'1'10".

Any partiallUm series can be written partialaum of the above aeries (01 - CIt) a.) (41

+ (aa (ae +

a.)

a,. a.-.1

a. -- a,.

+ ··· . . . + {{
nia even if niB if n is odd,

which is. ie a Bum aum of nonnegative numbers, hence nonnegative, and this partial 10m sum can a1lo also be written a.) is odd -
which is Bum ie is located in (0, G~. a~. If we ie at moet Gl. 01. That ii, ie, each partial aum delete the firat series we are left withplua or filSt few terms of an alternating alternatinl aeriea minua integers n > m we have min1l8 another UlOther altemating alternatinl series, aeriea, 80 for any positive integelS

1£, + - ... =a,.I~,,-+ ca.1 ~ llil (-l)f-~I-IG.o+l (-1)..-""1- "-+.-"-+1+"-+1 - G.o+t

1

G.o+a - • o. ::I::

G.o+'o••

t~ (-I)"-lea,. converpa. Each partial aum

-

=

Since 8m a. - 0, the aeriea M_ Um .. ....

.-1

(-1)"-10. convell-.

partial Bum

Ii. Beries ie is not sero since U. in t.he the c10aecl cbecl interval [0, (0, Gal. G~. The aum of the aeriea putial IUUlI IUma (al (G. - a.) + (0. (ae -- a.) 04) + + ··· ... + the partial + (0.-1 (0111-1 - a...) a..) are increuincreasing Bum ie is Ieee less than Gl a1 aince since it equala equals iDe and IIId positive for larp large ft, and the sum At-(o.-ae+a.-Clt+ fit - (CIt - CIa + tit - fit + ... ...))
• E -1)--'.. E E (0, Gs) aa).• ..... ((-I)-lea,. a-l

tbM tlUl thia reault impli. the aeemincl1 Note that aeemilll1y atronger stronger reault result that the diI..- heW_ . dilelWlce between the sum of the aeri.

Gs-Gt+aa-a.+ .,. CIa -o.+ae -a.+'" ft- partial aum is leas Ieee than ""+. and ita ft" partialaum 0.+1 in abeolute absolute value, aince since this difference ie apin an altematilll alternating series. aeriea. is properties of 01 absolutely abeolutely converpnt The main propertiee convergent aeriea, series, proved in the next two propositiona, are that their terms may be rearranged in any order Dext ftIl'Oupecl in any way without affecting or feIIOuped aflecting the converpnce convergence or the 8ums sums of aeri•• This mak. makes it poI8ible perfonn many kinde the seri•• poesible to perform kinds of manipulations Dl&nipulatioDa these seri. aeri. without concem concern about converpnce with theIe convergence problemll, problen18, a& fact that does Dot not hold for series aeri. that are conversent converpnt but not absolutely convergent dOlI cOllvergent (cl. (cf. Problem 14). I').

ProIMMJtlon. Let /: 11,2, t1, 2, 3, ... •.. )t --+ 11,2,3, (I, 2, 3, ... tJ be a function that ia i, PropoRdon. 1M

•

OJIHU Gftd ad cmIo. 01lI0. TAm Tlaen if E a,. ia an abaolulelll one-one a. i8 abaolutely convergent m'iu Beriu of real

....".,." 1M""" IAe __ fttItftlHn,

...-

..... a-I

E• CI/C,,) G".., U ia ,.,. abaolulelll • absolutelJl convergent convergent and ..... E• a/Ca) - E• CIa.

...

--I

12. Ilfnlflftl _

1'7

For any nUDlbe1'8 /(1)./(2) /(1),/(2),•...• .. . ,/("') AllY positive integer ft, ra. the numben /(ra) are a& subset 8um of the aeries series IRlbeet of 1, 2, ... •.. , N, for some lOme N. Thus any partial lum

t.-1-Ia" I. the latter aeries converges ita partial IUmB are bounded, hence al80 the t. I

I: I01(.) I is 1888 than or equal to a partial lum of the aeries _-I L I0.1. Since _-I .-t (I/CIa)

I is lesa than or equal to a partial 8um of the series

Since

the latter series converges ita partial sums are bounded, hence also the partial sums IUmB of the series aeries

L• IO/(Ia) Thus the aeri88 series L L• a"a) O/(Ia) I: a"a) I are bounded. ThUl •

•

.-t

--' a-I

.-1

is absolutely abaolutely convergent_ convergent. We know that

- -

. tD

CD

CD

la-I

.-1

L O/(a) - L a. == L (a/(.) -L 01(.) La''' (0/(.) _-I ....

__I la-I

0.) a.)

aba1l complete the proof by showing Ihowing that the latter 8um and we shall surn is zero. For any t > 0 choose a positive integer N such 111 ~ N luch that whenever n > m ICI.o+a I+ 10.. 1000+11 we have 1a.+ll +11 + ·_. + 10.1 < t. Then choose N' luch such that all the numbers 1,2, ... al11ong/(I),/(2), _... ,/(N'). ,,(N' ). Clearly •.. , N are included among /(1). 1(2), .. N' ~ N. If ft ra > N N'' we have

+

+ ... +

-

L

(0/(1) -

Gi) == ~ aa i~1

i-I

L

OJ, JESI

coats of thOle those integers!(1),/(2), intepns/(1),/(2), •. ..•,/(n) where 88 1 consists /(n) which do not occur amoDg consists of tru.e those integen integers 1, 2, .•. .. _,,n &mODI 1, 2, •.• ••• ,ft, , ra, while 8. coDBiBta ra which among 1(1), 1(2), . .• I(ra). Clearly 8a do not occur &010°1/(1),/(2),._ • • ,,/(n). 8 1 and 8. have no element any of the numbers 2•.•• in common and neither includes includ. allY nUDlben 1. 1,2, ... , N. N, 10 10 that. that B1 U VSaC I, N + 2, ... .•. ,Ml 8 8. C IN + 1, , M} for lOme M. ThUl ThUi for ra n > N' we have

+

Il!it

+

CIt)

(0/(,) (0/(.)

i-I

proV88 that This proves

IsSi~&VBtICltI ~ 10.1 s laN+l1 + I~N+.I + ... ·-. + 10.,1 IGJlI <

t_-I

a_I

ieliUs.

(0/(.) -

10N+ll

laN+11 .

e. to

a.) 0.) == O.

If 8 is a set Bet and cp: f(J: 8 S ---+ R a function then the expl'e88ion expression

tof cp(.) 'P(a)

~ .ES

well~efined in case C&8e 8 expl'e88iOIl can 8Ometim88 is well-defined S is finite. This expression sometimes be given a meanilll. independent of any ordering of 8, if 8 is infinite. In fact if 8Scan can meaning, be put in one-one correspondence with the natural numben numbers and if in 80 abaolutely convergent series aeries then we can define L cp(.) doing we obtain an absolutely \0(')

li .eB

to be the

BUm Bum

aeries. More precisely, if I: 2,3, ... 1-8 of that series. f: (I, (1,2,3, 1--+8 is a

function that is one-one onto and if -CD

L _-t _-I

t

.-1 _-t

cp(f(ra» f(J(J(n» is absolutely convergent. convergent,

then we define ~ cp(.) \0(') to be L cp(f(ra» \O(J(n» (which by the lut last proposition is ~

independent of the choice of I).

148

m.

INTERCHANGB or 01' LIMIT OI'BBATlON8 OPZRA'I10NS INTBBCIfANGB

Special scts 8S which can be put in one-one correRpecial cues CIUIeII or of infinite sets 11, 2, 3, ... }J are spondence with 11,2,3, (1) any infinite subset of the natural numbers (for the elementa elements of such 8uch a set can be written down in their natural order) (2) the set of all ordered pairs I (n, m) : ft, n, m -=- 1, I, 2, 3, ... .•• J of natural numbers (which can be written down in the order

(1,1), (2,1), (2,2), I), (1,4), (2,3), (3,2), (4, (4,1), (1, 1), (1,2), (2, 1), (1,3), (2, 2), (3, 1), 1), .•. ) (3) any infinite let numben set of disjoint nonempty subseta lIubsets of the natural natunJ numbers (which can be written down in the order of their ema1leet ImaU.t elemente). elemental.

following result sayB tenns of an absolutely CODVerpnt The followilll says that the terms converpnt aeries may be regrouped in any fashion without .,ttering eeriee ~terinl the abeolute absolute convergence sum. verpnce or the sum.

•

GO

Propoaition. Let I: L all Nria 0/ number. Proposition. a" be an absolutely ab80lutely convergent aeriel oj real "umber, II_I tl-l

, St, 8., ... and let 8 1,8.,8., . .. be a(J aequence sequence (Jinm (fl:nile or in.JiniU) infinite) 0/ 0/ dUJoint diajoint nonemply nonemptJl mad .,. tDhoae union 8 1 . .. iI it 1M fJI&tire ,ett 0/ natural number. whose V 8. 8, U V 8. U V··· entire .. ", 0/ oj 1U natural number' 11,2,3, (1, 2, 3, •.. ). for each i tuM such tIuJt tJ&tJt 8, it iI infiniU i",finiu 1M eM natwal number. J. Then lor Berm .~ ")' a.. it ill abtolutely absolutely convergent, if iJ the number 0/ . aet8 8" 8., 8" ... . .• it feria . 8B1,1, Ss,

tl~i

infinite then U&e.me. the 8eries

t (.et r 0..) a..) itia ab80lutely _II cme I: tJb80lute l1l convergent, twl - in ift (lftfl C~ 0.) '(_I.~ -1ft. . (.tf, a..) - ~ 0.. a... _1 ...1

ctJIe

neSt •••

For any infinite subset lubset 8 01 t1, 2, 3, ... J, ordered in a l8QUenae lequence in of 11,2,3, any fashion, l..bloB, each partial sum of the series aeries ria..l ~ IGal is 1_ 1. . than or equal to

...

L

.~

__I lome partial sum Bum of the series Beries I: 10,.1. some Ia..1. Since the partial 8UIJl8 IUms of the latter "-1 series are bounded, so r I0..1. Ia..1. Thus I: a.. 80 are the partial sums of I: t.Ia is i8 absoabeo.. ~ .. eB "eB "eB lutely 8 C (1,2,3, ... J).• lute1y convergent. Thus I: a,. a.. makes sense for any subset Bell, 2, 3, ••.

L

L

•nEB eB

We claim that

This is clear if eitHer 8Bl1 or 8. U B. . " · is a finite set. let. On the other hand S" V S. U V·· if both B1 B. U .. , are infinite then we can order them into 8 1 and 8, V B. S. U V··· sequences and then use part (1) of the second proposition of this section to get the same result. Thus, by repeated application of this idea,

12. I"nlll'l'll IlfnJCml.... una.

_ ... IIII~a.+~Gwt+···+~a.+ -···-~a.+ra.+···+ra.+

.u .EL

•eal.&It .U..EI.

~ ~

.......~.... .......

149 1.

.. a.

in case sets 8., 8 1, 8., Sa, Ba, •••. We are clone, done, ~ GCeP' cue there happen to be at least" least " leta in the case where the number of sets 8 1, Sa, Bt, Sa, 8" •. •• .. is infinite, infinite. where wbeN i' eeta 8., remains to show that the eeries aeries and that its ita 8um lum is

.li

~ ((.~ a.) a.)

CODftl'pIlt. is abeolutely oonverpn'

t a.. To prove that it converpB eonverg. to the ~ IUIIl eum t a. ..

.-t • -1

....

it luffices IUftiCes to show Ihow that

r

lim

.tJI~... .... •..... E.*sVI~

a.-O a. - o..

To do this, for any e1 > 0 choose N lOCh IUCh tbat tW if " > chooee a positive intepr H m ~ N then 10.+11 mob that that, 10.+11 + 1"-+1' IGo.+.1 + ... + IIa.1 a.1 <. and then chooIe H' moh

+

11,2, ... NI C SIU ... U8N'. V SR'. If now" .. . ,,HI C 8811 U U8.U any partiallum partial 8WD of the infinite eeri. aeries value of any

~ abeolute abIolute > N' then the r a. (aldOl ~ .. (taking the .u...tJI..'IJ. ."*sVI.. ~ ....

,-

terms of this series to be in any fixed order at &11) Ia.I, tenns all) is at most. .~ 10.1,

lOme finite subset of IN I, H where 8' S' is some fN + 1, N + 2, H N + 3, •.. ••. 1, I, hence henee it ie iI aN+11 + I(IN+I' aN+t I + ·· ...· + IaM 1 1_ at most I(lR+11 I for IIOtne 80tne M > H, N, hence is ill. . than

+'

._. a.. Ea..

e. zero and I. Thus the above limit is indeed aero CD

to 1: a-I

tbat see that

!i~ (.~ a.) .~ a.) (

.

.

..... 1a.1,

CODV8IpI indeed OOIlvelpl •

thi. to the ablolutely absolutely convergent aeri-1: Applyinl this eeri. E

'Clal.

we

....

..1) is convergent. Since I.~0.0.11 SS.~ ._10.1 ~ (.~ 1 10.1) 10.1 for all aU

!: (

... , the comparlaon comparison teet teat .howe that ~ (.~ .~..) a.) isII &bIoIutel,. ii-I, - 1, 2, 3, •.. abIolut.e1, eonvqent. This completes eompletea the proof. convergent. For infinite series of real-valued functions on a metric apace Ipaee we have resulta, all immediate consequenCe8 the following relults, consequencel of the definitiODl definitioDi and resulta of preceding sections. results

•

CD

Propoaicion.. ProJHMition. IfNI" apat:e

Tit. ifl/i1&ite uriu .,,;u 1: I. TAB infinite E!. • -t ..-I

oJ ,...,..,.luetl 011• . .tfNIrie .... 0/ ~ /wtt:lWU ~•

eonver," uniJormlJI un(formlJ (f 01&1, E eonvergell if mad tmtl Oftl" Uaen >m In ~ H N Uam

(f 1& em integer N IUCA IIuJt t1uJt if fa

"-+1(1') 1/-+1(1')

Jor aU 1'E for oU fiE E.

ii, • > 0, ..... if. "",,., ,.". tift, • .... ..." ....

+I-+h) /.(,>1f < •e J....h) + ... + +/.(,)

111 III

m. 'fII.

IIftW8CIIAIfOB OPIUlAftON8 ~G. OF LOOT LUIIT 0 . . . .onON8

-

• E ........a

The infinite series aeries E f.. I. of real-valued functions on a metric space E ill&id ablOlutely if the series t/ra(p) illaid to COfWer,e C01IfHII'P Gbaolulely I:,I.(P) is absolutely convergent ..-I .-a

lor each p E B.

E... 1: E •

Corollary. it Gft infinils "" maw of ret&l-vGlued reaZ-uGlued luneliou lunctioM on a a mdric metric 0w0IIary. II E If.• .. em irVinit. ... 01 ....a

.,.,. E E and ."ace

•

c:onHrferat ..,.... a. Ga C07&Hrfenl ,me. 01 real numbfr. number. lucia aueh that I/.(P) If.(,) I1Sa. So.

a-l ....a

for aU aU ft, IIaen lAm lor all ,p E E B CIftd aM all

-

-

• f. C07&Hrfu tmd uniformly uniformlu.• E I. ~ tJbeoluUlll abIoluIay and ....a

...1

• E/ •.. ...s ....

Propoaldora. uni!orml1l COftV8r,mt aerUa 01 0/ continUOtU continUOUl realPropo8ldon. 1/ II E f. it aG uniforml'll ~ ""ia

.m ..

"'*

IJGlued lAm ill ita au", it a CJ continuou continuOUI function DGlutd ~ on Oft (J a metric apace E IIaen _E. OR E.

-

E•

Pro".ltlon. E R, aG < b, and E I. it a uniformly ~ cmwergent P . . . .'don. 1/ II (I, a,"b E I• .. .-a __ 01 0/ COfttiftUOUI COIItinUOtU reol-t1cJlued real-tIalutl ItmdilJu [a, b) IIaen am. Itmditml Oft (a, Uaen

.-1

r. (~/.

)(aI). -

~

r.

1.(aI)=.

propoaltlon. Let 11. /1, It, 0/ real-uGlued reaZ-ualued luneliom Junctions Oft on an PrOJHMlcf.oft. 1M II, II, la, ... .•. be a ~ 01 inWval U in ift R, eac:A Iuwi"" a contiftUOtU open inlerval each 1uJvi"" continuoua derivative. 8uppole Suppoae that tI&e the

•

inflnit. .,;u etmfIef'geI uniformly on U and tMt E U tI&e infi,nite ..,;" EN tl.' corwerge. tIaat lor lor lOmB Bome a aE the .-a

•

.,;u E /.(0) I.(a) .ma .-a

conHrfCI. cotaHrga.

Tiaen 1M tI&e ""ia Tlam aerie.

a-a

fwt,t;Wm em on U Gfttl aM /WUJlitm

-

• E I. ~ f. converges to a G difftll'eratiable diJfermti4ble •• a-I1

-

. )' -- .-aE/.'. ( E/. t/a)' tl.'. •_1 -1

....

I S. POWER SERIES. Let a, Ct, e., Cl, Ct, Ct, ••• be real numbe.... numbel'll. The aeries of real-valued functions onR

...1E-: •

+

c,.(aI - a)· .. - Co + el(:': c.(z Cl(Z -- a)

+ ea(:.: CI(Z --

ill oaIled eaUed a",..."';" a ,..,. _ _ (in powen powel'll of z - a). fa

a)1

+ ...

I13. I.

151 lSI

PO. . . 811111. I'OWIIJlIUl"

To avoid meIIIIY meI8Y circumlocutiolll, circUmlooutioDl, one a.lao also calla calls the above expreuion expression 80Ine a.. power aeries when z:e is not the identity function on R but rather lOme specific element of R. The first question about power series is for which :ex E R the series converges. test; the conv8l'les. Here are three examples, all verified by the ratio testj immediately following theorem 8888rta theBe examples are typical. userta that these ExAMPLB

1.

Ex.uIPLB E:Jwm.II 2.

...t.::...",

~_x· ~ converges for all :e zE R R.• conv8l'les .... ral

•

I: Z" 1::e-

converges if conv8f1e8

.-0 ~

ExAIIPLIJ E:Jwm.II 3.

Izi < 1, div8f1e8 diverges if l:el 12:1 ~ 1. l:el

• l: I s- converges if z ..; ~ 0, div8l'lee diverges for all other :e. z. 1: nft I:e-

... ~

Theorem. For ,eriea [lor a0 given ,wm power potM' -w

t c..(z c.(z - 0)· a)" one 01 0/ the lollofIJing follmDing it I: 0t&8

~ .-0

"",: ",.,:

(1) TIN..ua for oil all z E R. 2'A. ItIJf'ia COfWer,. ctlt&fJCf'fu oNolutsly takoluUly lor (I) TUre 2'. . GiItI e:riaCa 0G reol (') real number r > 0 ad BUM that the aeriu Nrie. corwergu convergea ab.oabBDlutslJl tJUJl Iz - a I < r and divergu diverge. lor aU :eZ auch BUCk that luUl, lor aU oil z E R IUCA ad tIaat

Iz --ol>r. Is 01 > r.

(8) The,eriea only if z - a. G. 2'Ae ItJf'iu COfWerg. conIHlr'fU mal" [lUf'tIwmore, lor em" '1 rl < r in coa CON (I), ('), or lor an arbitrary I'urtMrmor" for Gfty a,bitraru rl ERin CtJ46 C(U6 (l), (1), the for aU z ad auch that 1% a I ~ rl. '1. 1M CMWergmcc ~ i8 ill uniform lor Iz - al

........-

aeries converges for z .. For suppose that the series a:

~,

for lOme some

E a,

~ ,-E ~

GO

and let 0

< b < If I~ - aal.I.

We aha1l E c..(z shall show that I: c.(x - a)a)A converges

abeolutely and uniformly for all z such that Iz - a 1I S b. To do tbia, abIolute1y this, note

•

eince ~ ~ c.(E c..(~ - 0)" a)· conV8f1e8 c..(~ - a)that since converps ~e ~e have lim c.(E a)a -== 0, 80 that

there exists a=-number

- a)· I S-- for

::;

,,-.cD

a number M such that Ic.(E c..(~ - a)" J

M for all n. R. If Iz ~

- a I S:s; b

then

1c..(z-a)·I"Ic..(~-a)-I·1 Ic.(:r:-a)-I-Ic.(f-o)-I-\ But

...t ~

,-

~=: I- SM-I ~M·I ~~a f~O

bl(f :- (I) a) I" I- is a leometric pometric ~riea MI b/(f ~riee with ratio Ibl(f b/(E -

1-·

,-.

a) (I) 1 I < 1, 80

t c..(z - converges absolutely and .... uniformly for all z such that Ix Iz --al S b. Now consider the set S of all

comparieon with this aeri. aeries by comparison

c.(2: - a)a)" converges absolutely and

~

uniformly for all

%

such

01

b. Now consider the

S of all

lSI 1St

t.....

VII. IHURCIIANGIJ INTIIIIICJL\NOII OF 0 .. LIMIT OPmtATlONS OPIlRATlONB

EE R

such that the seriesL series c,,(~ - a)" converges. It may happen that c,,(E .-0

B 8 == - (a), (ai, which is possibility po88ibiJity (3) above. It may happen that the set 8 is unbounded, in which case E 8 such that C88e for every '1 E E R there exists a EE <" case ', > r, -l.u.b. al:: EE 81. Then, > 0, the series diverp8 and for any ''1 1 <, '1 < IeIE - 01, ai, 10 that the < , there is a EE 8 such that '1 _riee 2: such that Iz IZ -- 01 GIS '1. series converpl converps absolutely and uniformly for all z ~ rt. Since '. W88 cue (2) obtains. obtailll. WM any number Jell Iell than ,r this proves that C8M

'1

'1

The number radiUl oj 0/ cmwergence convergence of the given number,r of case (2) is called the radiua interval oj 0/ convergence. In C&8eII cues power series, the interval (a - "r, a 0 + +,)r) the interlltll (1), (3) we also use the expression "radius of convergence", meaning by this the symbol •GO or the number lero zero respectively. aeries baa bu radius of convergence Ql), it may mayor If a power series convergence,r "'" 0, CIt, or may not converge at the extremities aB - "r, aa + r of the interval of convergence converpncejj for example the power series

+,

~ zt

z'

z' zI

z'-T+3-4"+ ... , zI-"2+3" -4"+ "', ~ zI ~ zI %-2+3-4+ ...• z-"2+3"-4"+ "', and

x+xl+r+ z+zI+zI+ ... all have interval of convergence (-1,1) (-1, 1) and the fil'Bt fint conveJ'les converges at both extremities, the second at one but not the other, and the third at neither extremitiee, extremity.

Lemma. Let

t c.(x c,,(x - a)" .... 0 .-0

0)" be a0 power W'ieB ,eries tDitIa with ,odiUll radius oj 0/ ~ contJergence

(pouibly r = 0 or ,r = III). Then the aeries r (poBBibly ,eries Q).

a)tl-l ....t nc.(z - a)-· nc,,(z -

tt~(~-a)·+l ~(z - a}·+1

.... n+l n+1 aho have luJve radius radiua 0/ oj convergence ,. olio r.

fil'Bt show that if the series We shall first

t c.(:z: c.(x ..... ,,-0

% x

a)· a)- converges for

= ~EP! pili a, 0, then the other two series converge for all == aU :z:x such that II:z:x - a I <

13. 13.

I,~E--

.

1'0"'''''' PO. . . . . . . . .

111 III

I. As in the proof of the last theorem, since 1: E• c.(E - a)· conVerpl conv. . .

..... ....

0 a ,.

the terms of this series approach zero, hence are bounded, a number M Ic.(E - 0)"1 M for all ft. ThUl ThUi 14 such that 1c.(E a)"1 S 14

10

I..!..=.!.\-I

exists there ate

\..!..=.!.\II-I

1 1 == nlc.(E n nlc.(f - 0)"1 _ a)"-'I= a)" I I.!...=!!.. I-I < •AM I.!..=.!..\-I n (:z:z -a I' c I, E ~ - 01 ~ - a0 _I Ec.. . . ( ) al E -- If IE -- al E - a• and similarly . tI-

),,+11 < MIE M IE - all.!.=..!.llI+l. 11.!.=..!..·\a+l _ a)-+II n+l E-a · II ~(:z: - _+1 f-a t ftC.(:z: - a),,-I and t c.+0.+ a

-C-"-( _

tI

%

-

ftC,,(x - 0)_-1 and 1 (z - .)..... a)1I+1 con,,-0 .... n ft. ..... .... eerie a I it therefore suffices to show that the series

To show that the series verge if II:z: x - a I < IE~-

•at

I-I IsI z -a -a1-'

nM AM

~ If f -a -0 IE -01 -AI E and

t

oll..!..=.!.\a+l

Mlf MIE - all.!.=..!.11I+1 .... n+l E-a ..... ~-a

.

converge, which is easily accoDlpliehed Uliog the ratio test. teet.. accomplished using Thus if

1: c.(:z: t c..(z _-.0

a)" has radius of convergence r, then either of the two

.-11

series obtained from this one by "differentiating term by term" or "integrating term by ternl" term" has radius of convergence at leut r. But the orilinal

..

..

....

~

1: c.(x - a)" can be obtained from series E fronl the series aeries 1: E tac.(a: tae.(z - a)·-I 0)·-1 by

.....

.~

,,-0

integrating term by term tern\ (except for the term teml for for"n - 0) and from the series

t

..... n

C+.. 1 C+"

a),,+1 by differentiating term (z - 0)"+1 ternl by term, tenn, eo 10 the previous previoue

,,-0 "

reverse, showing that the radius of convergence argument applies in revel'8e, eonv8llenee of the original series is at. least that of either of the two othen. Thus all three same radius of convergence r . series have the saule

.•

tI&e power ... wi-. c.(:z: - a)" Au . . 0/ Theorem. 11 the ia 1: E c,.(z AtJa ratl rad" 01 ~

....

(polftbly r = tI&e ItIftCtUm rr> > 0 (pouiblll == CD) tAen tAm eM Junction

IJ Oft on

+

(a - r, aII + r) (or Oft on •• R. V if

rr-CD)giHnbu ~ CD) giNn by

J(s) -

•

....E c.(z - a)·

u differentiable. diJlerentiGbk. FurtAermorelor Furtl&ermore lor Gnll flRY z E (a CD) it (0 - r, a ,. + + r) (or:z: (or z E ., R, Vr ilr - .) w1&mJe we have I'(z) .. /'(z) ==

nc..(:z: - a)"-' t..... nc,.(z

..

"

0)"-1

and tmd

J(t)dt.. t ~(z a).. J:/..• J(t). - .... ~ n ~ 1 (z -- a)II+'. _+1

+I.

1M

YD. m.

IMTIIBCIIANOS 01' or LIIIIT LlIIIT OPBBA.TlON8 OPSlU.TlON8 INTIDICDIANOB

By the lemma the three series involved have the same radius of convergence ,. Pick any positive number '1 < ,. r. Then each series converges [a - '1, a + rJl ,~ by the last theorem. By the 1ut last proposition unifonnly on (0 of the Jut laat eection section we have

-.... •_

(I: ( l: c.(s e:,.(z -

,_ , .

a)·) a)") -=-

..."...

I: l: (c.(z (e:,.(z -

-....

, GD ,CD a)a) a)") .. ... I: l: nc.(z nc,.(z - 0)"-1 a)"-1

(a - '1, a + rll. r~. Similarly the result on term-by-tenn term-by-term integration for on (0 [a - r" tJ a + ,tl r1l follows foOows from the immediate predecessor of the quoted z E [0 propoeition. Since rl wu was any positive number le11 Iell than" these same results resulta pIOpOIition. Since'l are Vue (0 -', - ',0 true on (a a + r).

'a, +

Ct, Ct, ••• ... be real numbers. We say that a real-valued function Let a, ee, Co, CI, aubeet of R Aaa Iaoa tAe the potDer potHr aerie. ezpanaion //on on an open subeet NrN' aponsion /(z) !(z)

-

• l: e:,.(z I: c.(z

a"

a)" a)-

....t

c,,(z - a)" there if

lor all aubeet. In this cue case I' /' exiata exista on the open IUblet subeet and has for aU z in the open subeet. power eeriee series expanaion expansion there, and similarly for 1", /"', etc. In fact, from •a pow. /", J"', /(z) - Ct Ce /(.)

+ c.(s Ct(z -

a)

+ Ct(z -

a)1

+ ea(z c.(z -

+

+ ... ···

c,(z - a)' a)1 + c.(~ a)· +

foUOWI foUcnn

+ 201(. 2oa(z - .) a) + 3ca(s 3ct(z - .). a)· + 4: 4 c.(:r; c.(z - a)1 a)' + ..., "', I"(s) 20a + 2 • 3ct(s 3ct(z - a) 4 c.(:r: c.{z - a)1 + ... /"(s) - 2ct 0) + 3 • "' ···,, /"'(s) c.(z - a) a) + ... I"'(s) -- 2 • 3ct ac. + 2 • 3 • "'4 c.(s ··.,, /'(.) /'(z) - CI OJ

/(·)(z) - "Ie. P")(z) ftle:,. + .... .. '.

For s - a (.-uming (assuming this point to be in the open set on which I/ i. is defined) we pt !(a) /(a) - ee,!'(G) ee,1'(a) - CI,/"(G) cl'/"(a) - 2I>"I'''(a) 2Ct,/"'(a) .. - 2 • &a, ac., ... 1',,) (a) - nlea. ftle:,.. .e ... ,,I<·)(a) reetate d1eIe these reeultlu reeulteu folloWl. follows. We Nltate Corollary. 1/ the fu'nctitm C0r0lIary. 1/"" /VI&ditm / Iaoa Aaa the

potHr potDer

•

...."t..

aeriu ezpanaion c,.(z - a)" ..,iea eqanaion I: c.(2:

Oft ... 0JHft . _ _ e1aat COftIoiftI ccmtaim a, lAm then 1 /Iaoa ccmtiftuou, tkrwotWA derivativu 0/ aU OR ... open .. , 0/ R UwJl Au con'inuoua .,.,., . . aM In particular, i/ if IJ hal 0If'dtn ""lI&it Oft lAw open 0JHft . aubM and c.. e:,. -_/(a)(a)/nl/or p")(a)/ftl/or aU ft. ft. 1ft 1uu JJOtM' ...... apataaicm in power. o/:r; 0/ z - Ga lAm then "'" tA.. power potHr aerie... • " .., . . . . apanaicm i. Uftique. unique.

.erie.

13.

10".. 8IIBmI I'OWD"'"

lSI

For any real-valued function 1 I defined on an open aubeet 8ubset of R that contains a, the power aeriee aeries containa a and possessing poIIIIe8Bing derivatives of all orders at a.

t

• I'->(G) n.·)(o) (31 - a)· --(z a)" _E _J ftl

".. AI is called all at 1M lAB point a. o. ealled the Taylor TG?J1or seriea tItIf'iu 011 EXAIIPLI1. 'Ex.ulPLII.

z) .. %+ Z' If 1%1 1311 < 1 then 1/(1 + 31) - 1 - 31

zI+ ..· '. There31'+ e e.

fore

Jog I og((11

-~ d t .... · + 31)" z) == J.{a +-t == %31 -o 1 l+t

l

sI 2

-Z'

%' - ~ + -31' + ... iftil 1311 < 1 -~ 1. 3 4 + ··· If % ·

+

Is %) valid for other values of Ie this power aeries lleriee representation of log (1 + 31) z1 diverges (and furthermore 311 Certainly not for 1%1> 1, 1. for then the series aeries divergee 31) is not defined for z < -1). Certainly not if z:E .. log (1 + s) == -1. -1, for the 8&Dle reasons. reuoD8. But if z z .. aeries converges. Does it converge to log 21 same - 1 the series The answer anawer is 1/68, flU. that is, is. it is true that

+

log 2

111 -1-2'+3'-4'+ ....

but this statement needs proof. Since a uniformly convergent aeries series of c0ncontinuous functions functiona has baa a continuous Bum 8um and the function log (1 +:1:) %) is

z -- 1. 8uffices to show that the aeriee continuous at % I, it suffices aeries

-

+

tL (-I)--lz"/ft (-l)lI-tz·/n is ..... .-1

uniformly convergent for z E (0, sum of any (0. 1). This is true since the 8um coDleCutive terms starting 8tarting with the ftlA number of consecutive n" baa has abeolute absolute value at mOlt a1temating Bee. series. most z"/A za/ft S l/n, l/ft. since for 0 < z S 1 we have an alternating

I is a&real-valued function on an open interval in R Suppose now that 1 containing CI a and that I1 baa derivativee of all ordem. containiol haa derivativea ordera. When doeel does f have & a series expansion in powers powem of % power aeries z -- a1 01 That is. is, when is it true that

--..

I(z) .. - f:,J<")(a)(~ E/(")(a)(z - a)-/nl? J(~) a)"/nl? "

Reverting to a& previous notation (end of

ChapterV). Chapter V),

I(z) -/(0) _ I(G) /(s)

+ l'(a)~1 -

a) 0)

+ ... + r·)(CI)~ r·)(a)~ -

a)0)·

..(z. a) + RR.(s, 0),•

•

....

we see that we have I(z) - E/<·)(a)(z EI'·)(a)(z - a)-/nl a)"/nl for &ny any particular zs

-....

if and only if lim B,,(:I:, R.(z. 0) a) - O. This can be & a useful criterion. criterion, 8ince since 'raylor'8 Taylor's

R..(z, a). theorem gives us some practical information on 011 R..

156

VII. INTBRCRANOJ: OPERATIONS INTIlRCRANOII 01' LIMIT OPIIRATION.

... :E ,,-0 CD

EXAMPLE.

The Taylor seriee series of e" f!' at the point 0 is E ~ z"/nl. This /n I. Tbie

.....

series R.. (~, 0) seriell converges for all 2:x E R. By Taylor's theorem we can write R.(~, ,tzta+t/(n e- is an ,,%,*I/(n + 1) I , where Eis some lOme number between 0 and z. Since e" increasing function we have

+

.'8' .-+1+1 • ~ 2:

Since

... :E E• z"/nl

.....

....

and therefore

IR.. (z, 0) IS ,,+ 1)1 IR.(z,O)I~ n+l)1 converges, converps, lim :I:,,+I/(n z·+I/(n

"...

....

...

1) I .. o. + 1)1 - O.

R.. (z, 0) - 0 Thus lim R.(z,

....

"

a40t

z" e-=-:Ee"" Eft-o nl ..... CD

~

for all E R. alI 2:zE ,4. THE TRIGONOMETRIC FUNCI'lONS. FUNCfIONS. 14.

We want to define the" the trigonometric functions and derive their standrigoroUs manner. The usual geometric way of doing ard properties in a rigorous ar~ length, relies on intuition, but it is possible to this, using angles and arll make this method entirely logical. However it is much simpler to use an altemate approach. We shall confine the discU88ion alternate discu88ion to the sine Bine and cosine 88 well as .. their functions, since all the other trigonometric functions, as inveneB, inverses, may be got from these. We look for real-valued functions on R that are everywhere twice differentiable and satisfy the differential equation

I" - -I. II:

II such a function I1 exists, from the equation /" If I" ... =- -I - / we deduce I'" == - -I', 80 that I is three times differentiable, then we get I(t) I'" let) .. == -I" ... == == I/ we get PI) 1(1) = == 1', /', I, 80 that IJ is four times differentiable. From I(t) "" PI) == "" I" / " == "" -I, 1(' 1) === -I', 1(1) j(1) - f', PI) f CI) === -/" "" == I, /, etc. Thus I has derivatives of orden and its Taylor series at the point 0 is all orders 1(0) + 1'(O)x I'(O)z - 1(0) zI /(0) + %' -_ 1'(0) z' xl + + 1(0) /(0) z' z4 + + ... . 31 41 21 •

z ;w! po! 0, Taylor's theorem givee For any particular % gives us the estimate IC.+I)(E) left+u(E) R,,(z, 0) -=- (n + 1)fZ,,+1 i)l2:"+1

+

lOme E for some ~ between 0 and z. x. Letting M be an uppet upper bound for 1/(E) I/(E) I, I, I/'(E) I for E ranging over the closed interval with extremities 0 and z, ~, we have

14....... '1'81001I0. . . .0 I'UIIG'IIOIII ",MCftO.. 14 OOIIOIIaftIO

.-....

Thus lim R.(z, 0) -

III:

111 III

_red

o. As A. a consequence, if a function 1 / with the deeired

properties exists, leriee for all z E R. exista, it is equal to ita Taylor 18ri. For any Cl, leries el, Ct es E R the 18ri. c"zI e,;t:' c,;et e,;eI 0tZ' ~

Y-ar +""ir + ...

+

CI+~-2T-ar+,,+ Ca f:tZ -

-

converps IfJeIl _ converpl!l for all sz E ft, R, U will be . . . by comparieon with the . __

.......

cll+lc.l) (l(Iell E• Izl·/nl. Taking es c.-'O,es-l we are + lesl) 1: Islw/nl. Takinl - 0, es - 1 we

led to de&oe define the

Bine sine function lin ain by

zI:rt' 11' z'

.•

81n2: -- z Z -31+51 I!I1nz - 31 +51

~7 Z7

--71+ 71 + ...

and taking Ct =cosine function takiDl Cl el .. - 1, I, es - 0 we define the coaine

COl

by

sI ~ s' zI zt 11' z -1- 1 - 2i 21 + 41 - 61 + ....

C08Z COl

The functions functiona sin ain and

COB

Beries series term by tenn term gives lives Ii . d:e lins mn z ~

are defined on all of R. Ditrerentia\iDl Dilerentiatilll their

Ii . di' COB z -- -1Ul z. :., ~ COBS -tins.

COl z, COBS,

Thua -I and any lunetlon fuDCtioD Thus lin ain and cos COB both .tiafy satisfy tho equation I" - -/ that satisfies this equation must mUit be of the form

I: /: R -+ - R

/(z) - C1COlZ elCOU +e,linz +e,lins

conatanta el, for certain coDstanta Cl, es E R. It follows immediately from the l8ries seriel expanaiona that sinO ainO -== 0,

COlO coeO

-1 - 1

Bin -ain z, COl COl (-z) (-%) - COl COl z. s. ain (-z) -== -sin

:zts

-k

+ 2 COl COB s';' S:. COl. COB S - 0, 10 that ainl s is constant. &inIO + _ _ 00 - 1 we ... .ain' z + coat COB's conatant. Since aiD'O t Bin coal zain'ss + coeI. - 1. formulae, fix lOme IOJD8 a To derive the familiar addition fonnulu, • e R. TbeD 'l1left . ain (s + a) - COB(S COI(S +a) : (z+ sin (s + a) - COI(z+a) COB(S + «) (ain'' sz (lin

+ coal coe' s) z) -

aiD sz 2211in lin s :. sin

!:s

and

!:s

COl (s (z COB

!

a) = + eI)

-ain (s (z + eI):S a)~ (s (z -sin

+

+ a) eI) -

-aiD -sin (s

+ a), «).

111

o.

'f'II. IIft'8IICBAJIOJI 01' LbIIT OPllaATlON8 m. IIft'8BCIIANGII LIIIIT OP&BA.TlON8

eothat IOthat (jIsin (z dI sin d:zI (x + a) == dzI =

+

. (

-810 -SID Z X

+ a.)

Hence we can write lin (x (2: + 0) COB:a:X + esain a) - Cl ClOO8 Cs sin z x for Certain Ca, Cl, Ct Cs E R. Differentiating gives COl COlI (:.: (:e

+ ex) s + Ct z. a) - -Ca -c. ain sin:e Cs COl con.

Setting lut two equations gives Cl Betting s:e - 0 in the 1ut

-

sin a, Ct Cs -

aI

COl 008 a, 80

that

+

sin ... ain:a: sin:e coea COlt a + con Bin (:e (:.: + a) a: COl z sin a COl COB

(s (z + a) =: ....

cos,

COl Z COB z 008, a

- sin z sin a.

To derive the periodicity properti. properties ~f tin sin and COl COB reuon reason as followa: Iin:es > 0 if :e since then all the exprelliODl expreaaions lin z E (0, 2), sinee :r;1 s'

:r;1 zI

Z7

:r;1 %'

Zll Zl1

:e -'31' 51-71' 5i -71' DI-TII' s-SI' DI-fi' ... are positive. Since tl COl :e/a z/tU; - -sin Z, d COlI (0,2). Now

COI:r; 008 Z

is a decreasing function on

11 1--+ 1 ... >0 1 1 0011-1--+00II1-1--+---+···>0 21

41

61

'

while

(2' 2')

2' 21 2' 21 2t ~ 0012-1--+-- ... <1--+-<0 00II2-1--+-41 41 21 4:1 61 - 81 -···<1--+-<0 21 4:1 ·. COII:e is sero aero at lOme eome unique point of the interval (1,2). It followa folio. that COlIS (1, 2). This unique point we denote ,,/2 or/2 (this is a definition of or;; note that at the moment we have only the rough approximation 2 < or < 4). We deduce that on the interval (0, COB % decreases from 1 to o. [0, .../2) or/2) COB:e O. Since the derivative of ains sin z is COBS, COII:e, which is positive if z E (0,11"/2), (0, or/2), sinz sin z increases on this facts that ain Bin 0 == .... 0 and sinll z + cosll X == 1, we see \his interval. Using the facta linx incre&8el from 0 to 1 on the interval [0, or/2). The addition that lin z increaeee (0, 11'"/2]. formulu live formulas then sive ain (z (:e +; +; lin

)-

+ ;) - - Bin sin z. +;) (z + ...) or) -= .... sin «x + 11'"/2) or/2) + 1t/2) or/2) == Repeated application of these give sin lin (x ain «z IGI(. (z + ...) .. ~ + 'r/2)or/2)- -lins, -ainz, Bin sin (s (z + 2'1')2or)- sin «z «x + 11") or) + 11") or) ... - sin Bin (x 11") -= fonnula, we get cos COB (x 2or) === • a. Similady, Similarly, or by differentiating the last fomlula, (z + 2r) COB:e, COl z.

COB COl

(z

aa -

OOIa. t. other words, ain and COl 211'". _:e. Ia worda, sin cos have period 2or.

15.

DIn'mIDft4ftOIC . . 'II1II ......a.u. liON DII'J'IUUIN'I'IAftOIf UND VIfDU 'I'IIIlIIftSOIlAL

119

,s. '5. DIFFERENTIATION UNDER THE INTEGRAL SIGN. The result we give here is only the simplest of a number of similar aIao the most useful uaeful and ita proof is illustrative of the resulta, but it is also others. otherB. The notion of partial derivative enters, but only for convenien98 convenien~ of notation. None of the properties of partial ditterentiation differentiation to be developed eseentially a in Chapter IX will be used here, only the definition, which is essentially one-variable matter. aubeet of JlI BI with the property that for each s E R the Let U be a subeet aubeet of R given by 11/ 1'1/ E R : (s, '1/) E UI ia the sublet (s,1/) Ut is open. That ia, ie, U ie union of open subseta aubeeta of vertica.l vertical lines /II. Then if IJ ia linea in the plane JlI. ie a realvalued function on U and (So, (Zoo 1/0) 'I/o) E U, by !(So,vo) *(:ro,'I/.)

"0

1/ into J(St, I(s., '1/), we denote the derivative at 'I/o of the function sendinl sending 'V 1/), exiata ij that ie ia provided this derivative exista

lL(So 1/) 1/1) _ lim J(St, I(So, '1/) .N..(So V) --/(s.,1/.) J(So, V.) 811 ,...... V-Ve av' ..... '1/-1/. ' exists. If .!L (So, exiata for all (So, 1/0) YO) E U we have a realif this limit exista. (Sa, 1/e) 1/1) exista By av I

*

valued function on U whose value at each (Zoo 'I/o) E U is lL (:ro, (So, ".) (So, 1/.), Ve), and

a1/

we of coune couree denote this function by lL. 8J .

ay a1/

Theorem. Lee Let G, b, c, ddE d, tmtlllll tmd 1MJ be G continuotU reedE R, G < b, c < tl, contimAotu realttalwd oj /II E' given bv by lIGluecl ftmd,ion. juradMm em the tile aubM 01

8uypoee tIaat tAGt Su",..

Z

I(:e, 'y) E':: aG :S ~:e ~ b, c < 'V dt. (:e, 1/) E /II :e :S 1/ < tll.

.N.. eNCa and cmtl ay eNla

i. continUDU8 on em tAg tAu ad. aet. flam ~1&en 1M I1ae lunctitm Junclicm g c:ontinuoua

tl) -+ F: (c, d) - R tleJin«l defined bv by

Jor aU 1/y E lor

F(y) -

L'J(:e, ,,)d:e

F'(y) -

L' -f(:e, Z(:e, y)tl:e

1/)u

(c, d). tl).

tl), both IJ and al/ay For a fixed 1/ y E (c, d), aJlay are continuous functions of :e for :e E (CI, [a, b), 80 80 both integrals illtegrala in question exist. We have to show that

160

vn. VJ1.

INTERCHANGII 01' or LIMIT OPllRAftON8 OPllRATlON8 INTEBCBANOIJ

F'(yo) exists and is 88 as indirAted for eaeh eallh Yo E (c, d). So let Yo E (c, (e, d) be fi"(Yo) (''hOO8e numbers c', tl d' such that c < c' < '//0 yo < d' fixed. CJ"1l00se d! < d. Then the set

8 == = (x, 1/) E E JCI E' : x E E [tI, [a, b), 1/ E Ie', [e', tf)) d']1 8 11 E

a!/ ay is uniformly continuous is compact, 80 that the continuous function a//a,l 1/) E 8, (ZI, (Xl, 1/U 1/.) E E 8 on 8. Given any I!E > 0 choose •3 > 0 such that if (x, (2:,,,) l and V yil < •3 then v' (z (x - %1)· Xl)· + (y (1/ - 1/.)'

+

~ IZ I

(x, 1/) (s, fI) -

~ Z I< (Xl, 1/.) 1 (Sl, fit>

b

~ a' G•

Wemay assume 3 < min (Yo - c', e', d' tI' - 1101. Yol, Then if 1/ Wernay Ulume that a y E Rand I, (x,1/) 111 - rIol < 3 we have (:r, y) E 8 for any zx E [0, (a, b). If in addition to III -"01 - 1f.1 < 3 we have" have Y '" 110 Ifo then I,

-1/., <. I'M -

J.tf.t• .!i....(x, ~(% 1/e)dx I By a1/ , "., .. If.tJ.t• (/e (/(X, 1/) !(x, 1/0) ..I II) --/(x, 1/-1/0 1/-Yo

F(w) - 1'(1/0) '(1/0) 11 - Jlo II -1/0

'(Js

Z,

J.. (:~Z ::I1L'( =

(x,,,) (s,,,) -

a! (s, (x, 1/0)\'-1 -_ a/ 1/0) ~-I

ay a1/

rr

I

(x"ho) ~I :~ (s, ,hO) )t:c

x and 1/) is alw'ys alw.tYB between 1/ where" (which depends on both sand 11 and fie. (We have U8ed (% - :1:)1 1/e)· == used the mean value theorem.) But V v' (x x)' + (71 - 1/.)' I" -1/01 < I'll 11/-1/01 -y.1 < 3, 80 that

''I -1/.,

+ (" -

'J

I:

I

1 : (s,,,) (x,,,) - : (s, (z, 1/0) fie) I < b

Thua Thus

1 I

if III Iy -- 1101 yol <

F(1/) - F(1/o) '(Yo) F(y) 'II - 1/0 Y 110 a,1I Yo. Therefore 3,1/ P' "1/0.

'~-II~ < I.'f.••.!i....( ! (s,z, 1/O)tb •

.:a..

YO/fUi

_

t I!

"II "/I

F'(y \ - lim F
~ G0 •

'dx J.tf.t•• .!i....{x a/ (s, 1/.)., ay all OJ

,,8/

,

was to be shown.

PROBLEMS

1. Find a sequence of continuous continuou8 functions

-.............

and lim Iim!.(z) Jim/.(z) exist and are unequal.

.............

Jra: R -+ --+ R such liml.(z) Jft: 8Uch that lim IimJ ..(z)

no8LIIIIB

-

161

......... .....

2. If !: I: E' - 1(0,0) f (0, 0) IJ -+ /(a, ,,), - R, three limits we can consider are lim lim !(a, r), !(%, 1/), lim lim !(z, f/), and .... .... ....

for J(s, !(%. If) r) -

ro' m'

Um lim

(••• ~(e.1) (•••>-11.1)

!(%. r). Compute theee t.heee limit., limite, if they . exist, J(z, ,,). .,

+

and for J(s, !(%, r) - :~ :; ~.

.r:r.

I:r.

I:r.

functional.: (0, [0, 1) -. R that COD~ COIl~ to the 3. Find &a sequence aequence of continuous functiona/.: aero function and such that the sequence 11(.)., I.c.a>a. I.(a>a, ··· ... ErG aequence /.(s)_, /.<->a, Ja<->a, increases increaaea without bound. functional.: (0,1) (0, 1) .... 4. Find a uniformly convergent sequence of differentiable fuDctione/.: -. R lOch that the sequence/,',/t',!.', such sequence fa', N, "', ... ... does not. not OODVerp. converge. S. ConBtmct a convergent sequence eequence of Riemann intearable real-valued fuactloaa 5. Construct functicma on (0, [0, 1) whose limit function i. ia not Riemann intearable. I) whOle 6. Prove the fonowinl fact. implicitly UI8d used leVeraI times ill in the tat: 1Por For aD)' &Dy 8. foltowinl 'aat, aevera1 tilllll

i::

m, a& aeries eeriea of real numben numbers ~ a,. 0DIy if positive intapr integer 1ft, CIa is CODVerpnt COIlV...... if and oDly

iJ ~ a..+. convergent, and in that cue .-1 0.+_ is coDverpnt, a-I

.-1 a-I

~a,.-tll'+41+ +a..+~a..+a ~CIa - tll'+ tit + ... +a. + ~a.....

f:t ~

.tf:t ~

Show that if til + tit 41 + til 41 + ... is •a convergent 88ri. eeriea of real Dumben numbers and 7. 8how &Del PI, Pa, ••• is. 1, 2, 3, •• " tIleD "1, 1'1, lit........ is a subeequence 01 of the eequenee IJ8qU8DCe 1.2. 3•.••• then (til 41 + ... + a..) a..) + (a.,.+1
+ ... +

+ ...

...

-~a,.. -~ Let til, tIJ, Ot, 41. 41, decreasing eequence sequence of poeiti.. poeitive numbers. Show that, that 8. !At til, .• .. •• be a decreasinl tit + 4J til + ... converges then lim ftG,. no. -- 0 (b)

til

+ tit 41 + CIt 41 + ...

-....

converges. converps. (InteRral test). teat). Let 9. (Integral

+ .. + 4a. 4aa + Sat 8aa + .,. ...

converpe converges if and ollly cmIy if fit tIJ + ..

J: 1% I: Iz E

.... R be •a dec.... decreuing poeitive-va1ued R : s% ~ 11 -+ lD1 poeitiv.. Y&1ued

fUDction. Prove that ~ !(ft) veri_ if aDd function. !(,,) con converges and ollly only if lim

1.. .....

/~)• .... /'·1(;1) .....

i::t i=t

(Hint: Draw a diagram.)

prececiinl problem to tell for which ,p 10. Use the preceding verge:

t ,,~,~. t

the followiDl following __ _ _ ooaCOIl> 0 t.he

,,(l~ lot" ~ lot a).' "(~ ,,).' ,,).' ~ ~"" loc" (~loc a)"

11. Show the convergence of the eerieI aerieI

tG-

to-a~_) ,.~a) of real·valued real-valued fUDctioDl functions on R -

I1-1. -1, -2, -2, -3, -3, ••• ... J.

12. Show that if til + lit 41 + lit converpnt .nee aiea of real tit + ... is an abeolutely abeoiutely CODVerpDt tI,1t + G/' G/' + ... eonverpa. converges. numbers, then Gl ~ + 01

+

13. (Root teat). Let

tt.. numben. Show that if there exiata exiatll a t .. hebe a ____ of real numbers. of

Dumber aU aufticiently sufticiently 1ar&e IUch that ~ S IIp for all large number IIp < 1 lOeb is abeolutely convergent. eoDVerpnt.

ft,

then the aeries

14. Prove that a aeries of real numbenwhich Dumbenwhich is eoDverpDt CODVerpnt but not DOt abeolutely eoDYe1pIlt such a way that the DeW new IIfIriu eerieI OODverpDt caD have ita terms rearraopd rearruaed in IUch oonverpB preusipecl DUJDber, 01' or IWIh such that the partial partialaulDl 008. . . . to any &DJ pree . . . . . . real . . . DWIlber, IUme of the Dew new aer1ea amaIl. eeries become arbitrarily 1afp, Iaip, or become arbitrarily ..n. converpot 1Illri. real numbera numbers t;t .. tt:; ..."" are ablolutely converpat eerill of ... then the aeries ~ a.P. &lao abeolutely Converpot, Converpnt, and ...... a.6. is &leo

15. tL Prove that if ~ .. and &Del

~aA-(~ ..)(~""} 18. nonneptive real numbers, let 8" 8., Bt, Bt, Sa, •.. 16. Let as, CIa, fit, Ga, lie, Gi, ••• be a sequence aequence of nonnegative'" be. infinite) of diajoint seta of natural numbers be • eequeDCe II8qUeDC8 (finite or iuDite) disjoint nonempty eeta wboee union is ((1,2, 1, 2, 3, ..• 1, J, and suppoae such that 8. & is infinite auppoee that for each i auch _ _ ~ .. eoDYeI'III and that if the number the eeriee a. eonveJ'l. Dumber of eeta Beta 8" 8 1, Bt, St, Sa, ... is

:m:

.=s, .:1'1,

a.)

1;; .. f=t

iDfiDite then the ~ (~ converps. Prove Prove that that the the eerill ~es ~ CIa the aeries aeriee ~ (~, ..) converps.

f:t ~,

CODverpL

17. !At Let V be a complete DOnned definibe. normed vector space (Prob. 22, Chap. III). The defiDitioaII of an iDfinite iDfiDite eeriea eeries of real numben &Del tiona and the converpnce convergence and IWD IUJn of IUcb aeneralile iJDJJMMliately aeri. of elementa elements of V. IW!h a __ eerieI ....... hmnecl.tely to eeries (a) Verify the ...... aoa1oI for _ _ of element. e1ementa of V of the OO8verpl108 oonverpnce criterioD of the flrat &rat propoIltion terion pl'OpOlition of 12. De8De the aotloa (b) Define notion of ablolute abiolutAa eoDverpIlCI8 COIlVerpnoe for __ of element. elementl of V and rearraDlinl and ADd fIII'OUpbac repwpiq propertill properti. of ..abIolutely oon&Del .-if)' verify the rearr&DIin& abIolutely coneerieI of ...... t. of V. Ye1pIlt IIrieI e1emeDte OO8verpDCe for a eeri. __ of V-valued fUG(0) Define the notion of uniform converpDCe fUDOtioaII OD on a metric apace aDd prove that the IUID tiona awn of a uniformly UDiformly COOV8l'pDt converpnt

.,...t

aeries of continuous COOtinuOUl V-valued funotioall eeriea functioDl ia is oontinuouL cont.inuoUL e., Cl, C" Ct, 18. Let Ce,

•••

.-

E R. Prove that if lim IC.!c..tll IC./Ca+ll exiata, exiate, it il is equal to the

radius of convergence of the pOwer .eeries ; . :~ ~ c. c,. zw. z-. Ce, Ch Cit Ct, ••• E R. Prove that the radius of converpnce 19. !At Let Ct, convergence of the power eeriea series

. ...

c.II ZZW is 1/lim aup ~. (Cf. ~ C (CI. Prob. 18, Chap. III for the definition of ~ , l/1im sup

11m IUP; IUPi the quoted expreMioD expn!IIIlon ia 8m is to be interpreted .. as 0 if the lim sup does aot alit exiat aDd &Del AI u •• if the lim tim IUP IUp ia DO' is 0.) ... J'iDcI I'iDcl the th. radli of CODVerpDce converpnce of the lollowiq •• followilll power .11: aeries: (a)

~ ftOoi n) .. ~.00I.)"

..,8L111U1 no.......

(b)

~ (loa ,,) ... - Z"

(c)

~ ,,-z;.

161

(d)~~ ~ (v'A)(e) ~~.

~ "I

21. Show that •a power eeriee aeriea ~ c" ~ Z" baa the .. me radiua of converpooe CODVerpnce .. :11. same Z", for any poeitive intepr integer .. m. ~ c.... e....., sa,

22. IAt Co, Cl, Let a, Ct, Cl, Ct, ••• E R, with at least leaat ODe of Ct, C1, Ct, ••• •• , DOnsero, Donaero, and let the

aeriee ~ c,,(z - 11)a)· have radius of convergence r. Show that power eeriee bave poaitive positive radiua exists a poeitive number 1 BUch that the sum BUm of the aeries aeriea is nODJero nonaero there exiata I < r such for ever,y every real number % Iz - Ga I < 8. z such BUch that 0 < F I.

23. Let a, c., Ct, e., Cl, Cs, Ct, ••• E R and let the power aeries ~ c,,(z - a)- have radiua radius of

CODVerpDC8 f(%) if Iz Is - at < r. Show that if 6b ER E .R and convergeDce ,r ~ 0 and converge to fez) 16 I" - aIat < r, then there exists a power aeries in powers of %z - 6b which conCODvergesto/(z) Verpi to f (z) whenever

lz-hI
«z -

in

a»·

(Hint: Expand out ~ c" «z - b) (Hw: ~) + (6 (~ -- a»- ~y the binomial theorem.)

24. Let G a E a, R, G" a" 0, 1,2, I, 2, •... .... Show that the "binomial eeri." aeries" 1 + a(a - 1)) ~ + a(a aea - l)(a 2) zI + ... 1+_ 0(11 21 ~fa lIZ 31 - 2} ~ ...

21

baa radius radiua 01 of COnverpDC8 convergence 1. Let f/ (s) (z) be the sum BUm of this aeries aeriee on ita interval of + s) /'(s) - qf(z), qf(s), and hence that fez) f(z) - (1 + s)· z) r(z) z)·

CODYa'pIlce. CODverpilce. Show that (1 Iz\ < 1. for ·114

26. Show that that. the eeriee aeries 25. 26. 28.

~ (n ; .,_-1 ....-1

abeolute1~ convergent and find lind ita sum. BUm• m) I is .beo1utel~

u. Problem 15 and the binomial theorem to show that (~t:) _~ (z y)" (2: + JI)" ((~ Z") (~~)

*' n! f:t nl

*t "I f:t ,,'

:I f:t

nl "I

z, II E R. Hence give an alternate development of the theory of the for &ll all s, exponential function ...

'11. Find. Find a real-valued function funotion on R r-ejDI poeeeeaing derivatives denvativea of all &ll orders whose whOle 'ZI. aeriee at a certain point coDvergee converges to the function only at that point. Taylor aeries (Hint: Start with Prob. 28, 26, Chap. VI.) (Hift,: 28. Define the functions tan, cot, derivativea. their derivatives.

880, 1IeC, C8C

in terms of sin ain and cos COB and compute

1M 166

VII. LIMIT OPIJRATlOlfl '9'11. INTlJBCBANOB DfTBBCIIANOB 0,. OJ' LDI1T OPIIJU.TlONI

29. Show that the functions Bin (ef. Prob. 28) are each increuing iDCreuing on aln and tan (el. (../2, ../2). the functions tan-I (on (-1.1) (-1. 1) and aDd R (-r/2. r/2). Hence define t.he funcUons mnaln-1I and tan-I respectively), differentiable, and compute their derifttivee. derivatives. reepectively). prove them differentiable.

30. Starting ~/tk (el. (ef. Prob. 29). 29), live give the TeUODIJ TeUODS 8Wting with the formula for d t&ntan-II s/u justifying the argument thaUor that for Izl I~' < 1 we have zI zI . tan-Isol+P· (1-P+14.... tan-I %1 : P - • (l - P + t4 - ... • •• ))",-z-3"+6"dt - % +

tf.- ", tf.-

and therefore

.

t- t - "',

.-1.. r

111

i- I -3+&-7+ ·• 3 + 5 - 7 + ......

31. (ef. Prob. 29). 29), mab make UIJ8 u. of the bi81. StartiDI 8tartiDc with the formula for dein-Iz/dz d ain-lz/u (el. nomiall8l'iea Beries for aIn-1 lin-I at the pobat point O. o. nomiall8riea (cf. (el. Prob. 24) to find the Taylor aeries 32. numbers and the oonverpn08 OODVerpnce and 82. The deftnitioDl defbdtioDIJ of an infinite series aeries of real numbel'll 10m lleriee of complex numbers Dumben (el. (ef. Prob. 111m of lUeh IUch a aeries eeri. extend verbatim to aeries 20, firR propoeition proposition 20. Chap. Qaap. III). Verify that the converpnce criterion of the &ra of 12, CODverpnce, and the I'eU'I'aDIiDI rearranaiDI and rearoupreproup12. the notion of abeolute oonverpDce. iDl _ries hold for aeriee eerieI of complex DUmnumiDs propertiee of abeolutely abeolutel7 convergent OODverpDt lUi. convergence extends to aeries series of complexben, and that the notion of uniform OODverpllC8 space,... 88 well .. as the theorem that the 111m IUDl of a valued functions funotiODlJ on a metric apace OODvergent series lUies of continuous functioDIJ uniformly convergent functions is continuous. continuoU8. (1bere (There is no need to prove any of this if you have done dODe Problem 17.) The notion of real power eeries aeries with complex coefticienu eoefBcienta in powen of a lUi. extends to power lUi. fim theorem of 13 paeraliaI pneraliul abnoet almoet vercomplex variable. Verify that the firat batim to such series. lOch complex power lUi •.

.

".

..

,.

".

(a) Verify that the complex power series aeri. 33. (a>

, " &' ... +i1+ij+3i+ 11 +ii+2j+3j+'"

".

" zt ,. 1-2j+ij-6j+'" l-ii+ii-iji+···

..&'

.. t:'

'-3j+5i- ... '-ij+6i-

e

eonverp for all. Prob. 32). converge all , E C (el. (of. Probe Denot.ing the t.he IOma ... 001 (b) Denoting mm8 of the earl. lleriee of part (a) by bye-, COl •• It and lin. aiD • reepeopreviowl conventionl tively (which agrees with our previous convention. if • E R). prove that

'1. e

,...e'I- ....... e-t."

e

t'I • ,.. -

It E C (cf. (el. Prob. 28) 26) and that for all 'a, ..

e

for aU, C. all. E c(c) Verify that (0)

,u-COI.+ilJin, "'-OOI.+ialn. aU +,.....

e'-+ ..... 008'---2001'---2-

.... . .---w". -- ..... r'·

81n%-~ am

cos' , + sin 00"'+ alnll •, -- 1 e C, c. and that the uaua1 for all ,• E usual equations hold for aIn(('1 .. + ..,. Bin + It).

001 COB ('1

+.., + It) and

no...... "BLaIS (d) Prove that any complex Dumber .,. 0 number ...

caD CloD

rec. tEC.

161

be writtea written • - II lor for lOIII8 lOme

(e) Prove that for any ,• E C and any positive iDtepr caD write • - "" til" int.epr • we ClaD for lOme lIOme II III E C.

aIpbra" atatel ...... 34. The "fundamental theorem 01 of alpbra" states that for ..., &IlJ poll,," poeitift in iDtepr" and any GI, exiata at leut mch that Gh Ot, III, ... , a. E C there exieta - ODe e C . . . thM

r

1+Otr-'+ ... +.-0. t"+Olrt"+G,r-'+IIIr-'+ +a.-o.

Expand the following outline ·into a proof 01 of dUe this theorem. (a) Let I: ..... C be defined by 1(.) . . 1/(1)'
+ .,-' .,-1 + ... +

lIn.

re

.a

J(,) -/er> - J(t) + G(. G(' - r)-(l (.r}f(.» 1(.) t)-(1 + (. - ",(.» ae in .. where m is a positive intepr, intepr,. E c, a .. 0, and &Del ,(.) iI • poIyDomial pol)'DOmialla (d) Chooee tJIA - -/er>/a -/(t)/- (el. (ef. Problem 88(e». 88(8». " Thea ChooIe aII e C aum IIIch that ttl" . . If II(t) (r) .. pi 0 we have II I for ..., any eufticieIltl)eufBcieatl1 .....u IID&1I poIlttft poeitift lJet lei),I < 'let)' real number I,t, which iI·. iI·a eontradictioD. contradictioD. Thus/(t) Thue! -- 0.o. (II, b) andand (e, dl be cloeed 35. Let (ll. Ie, d) closed intervals in ft R and let IJ be a continuous ..... rM1(z,,) (G, bl., valued function on I (z, ,,) E liJI: Et : z E (a, bl, 11 e E (C, Ie, dll. d) J. By Prob. 15, 16, Chap. continuous in" in , and L'/(z,,)dr II continuous !(z,,,)IIz i. continuoue !(z, ,). ia oonuuoue in ill ., z, 10 thM that VI, /"/(z.,)IIz

C,."

J.'

f

f."(f: I(z, ,).)dr J.·{/: !(z, ,,)dz). J:(f: !(z, /(z, ,).)dr 1:(1: ,,)dz).

and

£(1." !(z,,,)fly)dz I(z,,)dr). J:(1."

exiIt. Prove that theee intearals intecrale are equal by computiq exist. computiDI 1./. tJ/tM 01 of

for I E <0, (II, b). for' 36. Let fJ be a real-valued reaI·valued function

and

/"(!." l(z,,)fIr)J.' {J.' !(z, ,)fly).

aa. open opeD 111_' aubeet 08 .

:z {i!> (*D and ~ (t!) (i!) exist emt and are continuoue

of ... Pro.,. thM daM if 01 P. Proft

. , are equal. ('/" :z and and are continuous then . tbey hu been defined in the text; the definition 01 ......... ). (HWtI: has of 8/a. a/as iI isM. . . . .). (Hitat: u. ahow that if the . . I(z, Problem 35 to show (~t ,) w) e E .. .. :: z e E (a, lOt "I &It••r e Ie, eIIl fill II ellfile let on 08 which I/ is defined, theft tirely contained in the aet then

£(/.' :s(~).)'" :z(~)dr)'" - £(1.' J:(J." J:(J." :'(~"Ir)-.) ~(ID.)··) II, II, b, ce E R," ft, b < c, and let IJ be be. n.Iued fuaotiaD 37. Let 4, a continuous contiDuoua ..... real-Y&Iued functioa OIl the . . I«z, (z, ,) 11) E 19 (6, eJl. e] ,. Let ,: [6, el...... e) .... be ...... uaother fuDo&ioD. fuDotioa. W. We liJI : z ~ at a, ,11 E [6, F: 16,

-y that .y

I.+- !(z, /(z, If) ,) '" ". ....".. tmif"",." 10 F(r) J.+~ tmif.".", FCJI) .. [6, (6, el til If, for ..... > 0,

III:f:

Oft

I

there exiate exietll a number N e I(z, ,) •_ - 1'61) E R IIIch Ncb that !(z, FCJI) <. lor for II el (10 , E , > N and all ,rJ E (6, c) (80 that for each II e (6, c)e) the Improper iD.tepU iDtepal /'+-/(z, emte and equals equala FM (el. !(z, ,). If)dz exiete (ef. Prob. 28, Chap. VI». Pnm thM that if

J.+L+-!(z, /(z, !f)dz convergea uniformly to FM on [6, eJ. thea J1 II' iI J.+y)d% converges (b,cl, is 00Idima0ue. COIltinuoue.

U6 . .

m LDII'1' ""'1'10.8 OnaA'l'J0.8 We............ IJft'lIIICIIIAI 01' LDIn'

if _,6,e >0. if.,.,c>O.

II. Por" 2, ..... •.. let 1_ I. - /..11 a. SIIow Show that that. at. Par" - 0, 1, I, /.." lin- a c. , 4 • (COl • •_1.) (tl - 1) lin.... lins ..(a) (a) .. • (eoa • lID-I .) - (" aiD.... az - "ft . - :I:

"

,,-1 .-1

(b) 1_ I. --.-1 ~2 --.--1.... if. ih~2

(2!-l)"}

.. · (2a.•II. a.... -I 1)" (e) .. - 1l·a·6 4 ..... (Ja) (0) II.. (2ft) J j }

e

.,

lor" a, ... for " - 1,2, 1, 2, 8, •••

2· 4· .... (!!)

I...., _ 2 • .. • I ... (Ita) ''''''1 - a• a •6I • 17 ... ." (ta (2ft + 1) 1)

11, I., ... is is. decreasiDs eequenee aequenee havilll the limit 181'0 and (d) 1.. 1.. 1.,1 ..... a clecreuiDc I0IO IoIld lim .!ti.! I,"*' _ 1I .... I. (e) lim t • t • 4 .......... _ !.. w: • ', ,) 8m 2 • 2 • • 4 • 8 • 8 ••• (211) • (Ia) e ....11 • a 3• •a 3• • 6 • 6 • 7 ... ••. (ta (2ft - 1) • (ta (2a + 1) - 2 :I ((Wallia &I1ia product).

- ..

. . (a) Show dIU &11M if 40. U I:

-+.

,. e a ill oontiauoua, B :: •_ ~ 11II a iI ooatiauoua, then

/:*1 /(.,. t

t/(l') - AI/(s). + t/(i)

(J(t') (Jb)-

£+1 I/(.,.). (z)ds).

{.+lq. £+I

that. if , > 1 thea loi' _ q zC. Men (b) Show tba~ then . . ' -cliftera from - 1!2& l/ti by ... than 1/8&", (HiM: (HiN: Work out the iIltepoal __ for intepa1 uaia& WJiDl the T.,tOr TaylOr l8riea 101 (l (1 +.) at the point polo' 0.) 10& <e> U. pan (a)
1/.... + -)

(cI) (d)

.... .....

(.+1>

~00I.1- (.+~) ... +.) ~00I.1Ioi"+.)

u. (e) af of the ........ precediDc problem to aomput.e compute the above limit, t.hua thua u.. pan (8) obtaIDiDI obWaiDI tim lim

.~ "~ W'ft

-.....·r ... r-

I1 (8tirIiq'1 (8tirliDl'8 formula)• formula).

41. "or I'ar •_ e I_ - .il : , - 0, *1, ~I, *2, ~2, ... J (which is the E a, at .. let .,(.) .,(:J) - miD I ( Iz diItuae rna •• to the .DeanIt . . . . iDtepr). diduace flGlD iatepr). Show that . . . ill . a eoatiDuoua fUllOtioD I: a R --+ a R liven by (a) eoatiaUOUll ,UDOtioa

&

l(s) - ~ cp(lO"s)/tO" lea) tp(ICh)/I'" (b) if ...... _ IlUIDben DUlDben which have deoimaI • aDd , are ,.. decimal expaDIIioDi expaDIiODI which are equal.....,t tbeir .. ,. deeimaI ...... for lOme i > 0, . &heir ~ equal ..." iD ba their deciJDaI P placee, . ,. ," d '. . . . . by 1 .......... ~ fl'OlB 61, tbeo then .... ·beiDI diItIDot frolB the pair If. t~ II,

'01' ... ,

..<-) -

N) -- .-(a) - .10-' ~IO'"' fJ{J)

.........

16'

(0) if. .... II ia ill (b) ..... ill . ODe -.. 0, 1•.. 'J i - I tbaD then it. lAd, uad , .... . of . . ...... . . . . 0.1 ••..• i-I .,(1&7) - ,*10-', .,(10-,) - .,(10-.) f'
wbDe it • ~,. . .,(1",,) while if " thea ,<10-,> - .,(10-.) .<w-.) (d) if. aad , .... are .. in tbea it.ud' 1n (b), (b),taa. 1(,) 1<') -1(11) -lfII) -

t~ .Ut·" .10'"',

10 that -/(s»/ C/CJ) -I(;II»/CJ odd or eYeD odclor.... .. (.) tile .... aoatiD. _ ..... fUDlllloa/: a .... a .. (e) . fuaotioa I: ..... II DOWhtre DOWhere cWfereDt.iabIe. clilereDtiabie.

CHAPTER VIII V III

The Method of Successi"e Approximations exiltence In this thia chapter a number of important exiItence theorems are proved by a IUccellive auooellive approximation method. By way of introduction to IUCCleIIive suceellive approximatioDl, consider Newton'. Newton', method of IIOlvilll IOlvinl an equt.8quamations, tion/(z) Suppose that/. that/is a continuous nal-valued rea1-valued tion/(s) .. - 0: Suppoee attaina function on an open interval U in R and that 1 I attains the value RIO zero at lOme point of U. We wiab wish to find thiI this point. Assume I is differentiable on U and that I' Allume that I. is Aft) on U (10 (10 that I hu the • continuous and nowhere RIO sero at only one ODe point of U). Let s. Sf E U be IOID8 lOme value 181'0 first "the root of I(s) - O. Then lint appro~hnation app~ilnation to the I(z. small A. Since I(s. /(s. A) ia iI I(s. + A) - 0, for lOme mnaIl "approximately" 1(:zI) I(Zo) + A/'(:zI), A/'(Zo). by -tina .ttinl the latter expression zero we have A "approadmatelT' uapproximatelT' expl'ellion equal to RIO --/(z.)II'(z.). !(%e)//'(%o). Hence we let the nen appawimpoo approximation to the root of I(s) /(z) - 0 to be SI Sl - s. Se -/(~/I'~. -!<->//'(z.). (Geometrically, SI %1 •is the point of interaection intellection of the :z-axis tanpnt to the curve, at tbepoint s-axie with the tanlent curve II -/(s) -/(s).' (s.,/(:zI».) ptaa better approxi(%0, /(Se».) If SI z, E E U we can try to pt ~ - Sa mation to the root by I18ttiq lettilll s, s, -/(~/I'(~. - /(.dll'(~. s.. '1'b.ua, Thus, ptOYided pl'OYided If St E E U we can similarly define :z.. &at. . a IeqUeDGI we never l.ve leave the interval U, we . eequ8DG8 of pointe St, ••• of U such poin. St, Sl, ZI, St. IUch that 1(-,) . Sto+l Ztt+l -- S. :a:. --~, .. - 0, 0, 1,1, 1, 2, ~••••• ••

+

+

Ia'.

+

betW..,.,.....

r<.)'" -

If thia BeqUIIlC8 conv_ CODverpi to a poiat feU, . by this eequence point E E U, . daeD bJ continuity we haye

--Afr,

ff f-f-~\~l

I(E) -== 0 and fEis our deeired that !(E) desired root. lOOt. It.,.. It . . witJl.. withsaying that this tbia pmcedure -* ..,. out Dying procedure doeI doeIllM ahra1I work. 8everaI pc8bi1itH. poabilitiee are iDultnted Several illustrated in the fipte fipre _ OIl &be the next pap and only in cue do we arrive at • root.

80

1,. 1.

YIII•• ~ APPIlOXIIUTlO... ftD ••v ~ ~"'ON.

(a)

(b) Ploua 31. Variouleuea 01 Newton'. JDMhocl.

11. II. THE FIXED POINT THEOUM. THEOREM.

The lollowiDa theorem . , . that a certain rather pneral problem can Tbe followiaa I&1B

&I. . . be IOlved moat aimple-minded alwa:ra I01ved by _means of the moet timple-minded kind of auccaive eueee.ive appruimatioa. of this tbia chap_ neD chapter we approaimation. In the remainder 01 chapter and in the next ...... ... _ how thia.., tWa.., reeult rault can be applied to a variety of apecial probl8llll Iball special problema 01 CIOIIIlcI.able ClOIIIlcWabie moment.

n........ na-r.",.

com'"

1M II ......... ", . .".,. tMIrio IJICICC, F: E be ....crio .,.., II - E II • fuM, . IAtIft "'- , ~ - IItGt .. auc:A IAtJt lor aU all

"'"'" tlt,,,,,- li CiM. .",... s..,.... .... cw. __ • ,.., "umber

p,IEB .. ,.. ',fEB AGN

tl(p(P),p(9» ktl(p, f). 9). tl(F(P), F(f» S W(p,

2'_ ........ - IItGt f'AenI1we aitU • wituf unique poW P E B II . auc:A IAG& F(P) - P. FurlAermore purlAmnore if PI it . , poiftl B. ontl F
iI_"

-

If we apply the given inequality to dittinct points p, 9I of E &at II distinct pointe II we set il ~ O. If II II does not contain distinct that it is if II tinctB doeI dittinct pointe, points, tbat E conaiata of a linslpoint, the inequality holds U8UDle it given that holda for il - O. Thus we may IUIIUIJl8 OSi
,.., I'(p.), ,,- 0, 1, I, 2, .•.• ,,*1 -- P(p.), •••. For ...., J'GI' lIlT ....... Intepr " > 0 we we haw haft 4(,.,,..a) tl('(p...a), ,(p.» P.)· 4(p., Pa•.) - cI('
,1. • 1. rJUI).eII..., ftUD POI1ft' THO... TUOHY

111 I'll

Repeated application of this gives d(p,., ~ k d(P"-l, d(P,,_I, P.) ~ k k'l d(p._I, d(P"_,, P._I) P __t) ~ •. •..· d(p., P"+1) P..+l) S p,.) S P..-l) ~ klad(p.-a, 1'_-1) 80

that

d(p., P.+l) d(p", P"H) S ~ k-d(Po, k"d(Po, PI). PU.

It follow8 follows that for any integel'8 integers n

> m > 0 we have

d(P_H' P ..+2) + ·.. ... + d(P.-l, d(P,,_I, P.) p,,) + d(P_+I, P",+I) + k-+ k-+1 d(p., d(po, PI) + ·.. ... + k"k,,-I1tl(p., d(po, 'I) PU ~ d(po, Pl)(k- + k-+ k-+1 + k-+ k ..+t + ···) ... ) S

~ d(p., P_+l) P..H) d(p., P.) p.) S fll ~ k k" d(po, p.) PI) S

1

I

I

d(Po, PI)kPI)k" 1 -k ' l-k last 8tep step using the equation for the SUlll sum of a geolnetric geometric series. the Jut aeries. Since lim kk" -== 0, the sequence PI, Po, Pt, Pl, PI, . .. is a Cauchy sequence. E is com-

-.....

plete, 80 this, this sequence converges to a linlit, limit, say lI&y P. That is plete, P ... == limp lim , •. -.co "• .....

~ d(p, q) shows that F is uniformly conThe inequality d(F(p), F(q» S

tinuous, hence continuous. Thus . F(P) .. - lim F(Pa) F(p,.) == - lim P.+l p.+1 == = P.

.... ....

.........

show that P is the only point with the property that F(P) - P, auppose To abow P,8Uppoae that Q QE E H, E, F(Q) - Q. Then d(P, Q) - d(F(P), F(Q» S /ul(P, Q). ~ 1cd(P,

Since II; k

thia hnplies implies that d(P, Q) < 1 this

... 0, 80 that P =... Q. ==

of a metric space map if A map nlap F 01 lpace E into itself is called a contraction mop there exists a real nUlllber lcd(P, q) for number kA: < 1 such auch that d(F(p), F(q» S ~ 1cd(p, all p, q E E. A fized pea point for It'I" is a point PEE such that Ji'(P) Jt'(P) .. - P. The theoreln theorem .yl, 11&18, in brief, that a contraction D\&P map of & a nOllelllpty nonempty complete metric apace space baa a fixed point. The theoreln theorem alao also UBert8 aasertB that this fixed point is unique and it gives a 8huple lucceeaive approximation procedure simple auee_ve for finding the fixed point. If we check the details of the proof we see that we can even estimate the accuracy of any approximation approxinlation of the fixed point, consequence of the inequality for one direct eoneequence

d(Po, Pl)"pI)k4ft) < d(p., d(p P) d(p . , , ,-1-k -,rl-k Pi)id(...- P) < d(po, d(po, p.)kd(p•• 1 - k · ...., P) S I-A:

171 111

mt. 8t700U81\'11 8t7001188l\'11 APPBOXlIIA'ftONI APPJIOXlIUTlONI YItI.

Thus for any Po E E we have

dCPo, P) P) S tlUj:. d~, tlCPe,

!:
Proptnltlora. Let a, b E R, a < h, b, and aM let F: (a, b]-+ b) - [0, (a, b) be continuPropo.Jefon. lM a conti",,OUI/Ufldion. 8u'PfHI3e that F u ia diUermJioble differentiable on (a, b) and aM UuJt that lNre there ~. ~ oua/uftdion. 8UPPO'6 atJ ,etJl real number 1ck < 1 lUCk auch tho' tha' I F'(x) S kIe lor all z:a: E (a, b). TAM T1aen F it ia a F'(z) I :S for Gil contrtJdion trUlp, map, 10 3D UuJt that the liz«/' jiud point poim theorem i. ia applicable afYPlU:able to (a, b) aM contraction t:mtl F.

(,,onsists in showing that for all 'P, 1', q E (a, b) we have The proof of this (',onsists F(9)1I S Skip - F(q) Ie Ip - ql. ql. This is clear if l'p - q, whereas if 1''' p" q the mean value theorem gives us the existence of BODle some fE between pl' and ,q such that

IIF(p) F(p)

F(P) - F(q) ... == F'm(p F'(~)(p - q), f),

so 80 that

ql Skip skip IF(p) - F(q) I == IF'mllp IF'{~)llp - II

PI P. p, 7't

Ca) FlG111UD FlGtJU

p. 'Po

P.

PI

(b)

p,

ql. ql·

p.

,. P. P. PI

(e)

mape of a cloeed c10111!d iDterva1 interval in R. 32. The fixed point theorem for contraction mapa ia the graph of the function. In each diagram the curved line iI

provides a specifio epecific The important point about the proposition is that it provides. procedure for finding the point P E (a, b) such luch that F(P) P(P) - P, not that tens WI us that such it teD. luch a P exists. The mere existence of a fixed point P for GnU [a, b] any continuous map F: (a, [a, b) --+ - (a, bJ can be deduced from the intennediate intermediate (a, b] b) whose value theorem by noting that the real-valued function on [0, value at any zx is F(z) F(x) - x is continuous, nonnegative at a, and nonpoeitive so that it equals zero somewhere on (6, (a, b). at b, 80 The proposition can be used to solve equations of the form J{z) I(x) .... o. :II:

,2. 12.

rvMCI'ION8 IMPLICIT JVNCftON8

171

Suppose in fact that a, b E R, a < b, and that /: I: (a, [a, b] --. - R is a continuous function that changes sign on [a, b). b]. Suppose further that / is ditYerentiable differentiable K, E R 8uch such that 0 < K Kl1 S I'(~) f'(x) S K. on (a, b) and that there exist K Il , K. x E (0, (a, b). (ThU8 (Thus If is an increasing function on (a, [a, b); b]j changing / to for all % --If would enable us to handle decreasing functions that happen to satisfy _tiefy analogous conditions.) Then we can show that the propoeition, proposition, and hence the fixed point theorem, are applicable to (a,b] [a,b] and the function F(z) x -- cf(x), cl(x), where c is any constant such that 0 < c S 11K note % I/K•••. To do this Dote x E (a, b) the number F'(~) F'(x) first that F is continuous and that for any % cJ'(x) is at least 1 - eK. cK. and at most 1 - cK 1I , hence is nonnegative 1 - cf'(x) and less than some number less than one. In particular F is an increuinl increaainl I al80 also increases on (a, [a, b], I(a) < 0 < I(b) function. Since J bl, /(0) /(b) and for any x E [a, b] %

a


cJ(a) == F(a) S F(x) S F(b) :==I b - c/(h) cf(a) = c/(b)

< h,b,

that F F actually maps [a, b) b} into itself. Thus the conditione of the proposi[a, b] is 8uch such that E ~ -- F(E) tion are indeed satisfied. The fixed point EE (a, E ~ - cf(E), cJ(~), that is fm J(~) ... == 0, as desired. We remark finally that if we chooee c= some fixed %0 Xo E [a, b), b], set F(z) .. c/(x), and then try == 1I/,(Xo) 1/1'(~) for SOhle a: x - c/(z), Xl, %1, X" %a, X"~ ••• by the recursion relation Z"+1 XnH - F(z.) F(x ..) for fa n -- 0, 1, to define %1, 2, ... , we obtain a well-known simplification of Newton's Newton'. method which, however, does not always work.

80 so

,2. THE SIMPLEST CASE OF 12. THE IMPLICIT FUNCI'ION THEOREM. It often happens that for a given function I of two real variables we want to solve the equation /(z, y) = == 0 for 11 in tenns of z. That ie, for a I(x, 1/) given real-valued function IJ on a subset of E' we want to find a real-valued function tp I(J on a subset of R such that for all % X in the latter subset we have I(x, tp(x» l(J(x» == o. The problem 88 as thus posed is unwieldy. Among other diffiJ(x, culties, for a given real value of xX there may not exist any 'II1/ E R such luch that !(x, y) := any such numbers 1/, f(x, 1/) = 0, or there may exist III many II, the number of thetn them possibly depending on x. Even if there exists such a function cp(z) fP(x) there is means of Uexplicitly," that is by Dle&ns no reason to expect that it can be given "explicitly," some sort of fornlu}a, formula, 80 so that actually "solving" for 11 in terms tenna of z, x, or "finding" f(J, I(J, is literally out Ollt of the question. The lU08t most that we can hope to do, and this \vould would be of IIOme general ROU16 moment, nlOJnent, is to show that under certain leneral conditions there exists exist, somI' 80l\\~ function tp I(J satisfying so.tisfying the equation/(z, equation/ex, .,(~» fP(z» - 0 and possessing other dcsi I'able qualities, quali ties, sllch desirable such as being defined on a fairly being unique. To be 80Dlewhat large subset of H, c,,"tinuoU8, or heing somewhat more R, being ("~lltinuoU8, more pt'actical, Vl'Ilctical, "ge we refonnulate our problenl problem 88 follows: specific as well as &8 Illore We assunle fun(~tion !f is defined and continuous on a assume that a renl-Y\lued renl-nlued fUlletion we ask whether given open subset of E'!. 1~~ and "-e \vhether there exists a continuous reaIreal-

-

lOme nonempty open subset of R such that for all % valued function "f{J on some :e (.1:, f{J(%» in the latter subset the point (:e, ,,(:e» lies in the given open subset of gt E' and 1(%, I(x, f{J(%» ,,(x» - O. These conditions are not enough to 888ure assure the existence of a eolution; BOlution; for example if I is always positive then f{J(%) ,,(x) cannot be defined h) in the for any % x E R. We therefore suppose we are given a point (a, b) such that/(a, that I(a, b) h) =- 0 and we insist that the function given open subeet of gt E'such f{Jbe defined on some lOme open interval containing a and that "(0) f{J(a) - h. "be b. But z.I and (a, b) h) - (O, even this is not enough. For example, if 1(%, I(x, ,) rt) - zI (0, 0) then "f{J cannot be defined for any real number % :e ,. ~ O. The trouble in this lut laA aample example eeema to be that for any given .1:x near aa the function I has an extreme value at rt , - b. h. Thua we need some lOme condition guaranteeinl guaranteeing that for any given .1: x near a the function I actually goes both up and down as ,rt obvious way to do this is to suppose that aI/a, varies near h, b, and the obvioua aI/art exists near (a, b) h) and is continuous continuoua and different from zero. A. Aa a matter of fact this condition is sufficient for the solvability of our problem, as the shoWi. following implicit function theorem shoWl.

+"

na.....m. 1M r«&l-vaIUtd/unction /.4.'1 Theorem. La I be (IG continuoua continuous r«Jl-valtud lunction on an open ..."., auNet 01 ]I.'! IAGt contoi... contoiftl 1M point (a, b), h), tDitA with 1(0, I(a, b) h) - O. 8uppoae IJuJt aI/a, and 1110I SUPPOM t1uJt aI/a" eNta Gt&d ia eontiftUOUl on 1M , _ open tublet aublet and t1uJt IJuJt lL (a, h) U contiftuout b) ,. '" O. TAeft there tMre a, art

e:eiIC open ifttemJla u, VCR, tDitA E U and bE V, auch tAere uiIt. en,t, esiIC iaImIala U, with a E aucI& tAat that there a unique lunction I(:e, 1/1(%» ,,(x» .. a 0 lor all :e auch ftmclion ,,: f{J: U - V auch aucI& that 1(%, % E U, and aucI& IAGt thia continuow. t1uJt tAia It.mClion lunction "f{J ia it continuoua. We begin by defining another real-valued function F on the same open subset IUblet of EI HI on which I is defined by

F(:e, ,) y) _ , y_ _ F(%,

~jx, 11) y) !]%,

,,(a, h) ar(a, b) Thi. Fhas .. baaic properties that F and aF aFlay This F as basic /a" are continuoua, continuous, F(a, h) b) - b, (%, ,) :aF (a, b) - 0, and for any (:e, y) the equation 1(.1:, I(:e, y) .. 0 holds if and

a,

only if F(x, y) - ,. fl. The last F(.1:, ,) laat property indicates the main idea of the proof, which is a judicious application of the fixed point theorem. For this, we chooee some lOme r > 0 such luch that the open ball in HI JJ:I of center (a, b) and radius radius,r choose is entirely contained in the open set on which II is defined. Since aF iJF/iJ1I / By is continuoua and aF (a, b) - 0 we may assume auume ,r taken so small that continuous iJy By laF/a,,1 < 1/2 at each point of the ball. Choose k such that 0 < k <', < r, laF/a,1 then choose V r l -- kktl and such that IF(.1:, F(:e, h) b) :,:. ~. b chooee "1& such that 0 < h < V,I hiI < k/2 whenever Ix 01 < h, this last demand being justifiable by the Is - al continuity of F. We .hall prove the thcorem ... (0 - h, II 0 + IL) h) and theorem with U =

,2. 12.

v '"" (II(b c:

IIU'LICliT l'UlIaftON8 JULIa., I'UNC'ftON8

175 I'll

k, b h + k). Consider Conaider any fixed z~ E E U. For any II 11 E E R 8uch that

1Itly - bbl1~ k we have

d«~, 1/), 11), (4, (a, b» h») === V(z V(~ -d«z,

a)t (y - b)1 W < VAl V ht + leI kt 4)1 + (tl

(~, tI) y) is in our open ball of radius radiU8 r. If also that (x, then by 'the the mean value theorem we have

80

1/' E E R, 1/

<', Iy' - bl S ~ k, I'l

F(~, 1/) 11) - F(z, F(~, 11) 11') -=_ aF (z, (~, JI")(y y")(y - y') F(z, 8y 8'11 lOme 'Il' y" between 'II11 and 'II 11' (or, if JIy ... JI', y', for 1/' 11" - 11y :ra for some (~, rI') 11") is aleo also in our ball of radius radiU8 " 80 we deduce (z, II:

IIF(~, F(z, y) IF(~, tI) y) IF(z,

- F(z, F(~, y') y')1I S; ~

hi IF(~, 11) bl S~ IF(z, '1/)

:Ill

~ IIyy -- y'l· y'l·

+

F(~, b) 1+ I IF(z, IF(~, b) - bl - F{z,

Iek

1

11'). The point tI).

Iek

Iek

II1 -.bl +"2 ~"2+"2<"2 < T'tI-.bl+T:S 2" +2 - k. ThUi Thus the fixed point theorem is applicable to the cloeed interval [b - i, k, k] and the function that senda each 'IIy into F(z, F{z,1/), b Ie] 1/), a function that maps mapa this interval into itself. (Recall that zx is i8 fixed.) Thia This gives us the existence of a unique 11g such g - bblI S 8uch that I10 ~ Ie k and F(;,;, F(~, 1/) 11) == = 11, ,1, that is I/(z,l1) (z, 11) .... bI < Iek by the last displayed inequality; inequality i =- O. Notice that in fact 111 - bl that is, 11 E V. Since this is ia valid for each z~ E U our desired function tp is defined by tp(:r;) tp{z) .. g 11 and to complete the proof of the theorem it remains only to prove that tp f/J is continuous. But the continuity of tp can easily be allay is not deduced from what has been proved already. Note first that a/lay zero at any point of the open ball with center (a, b) and radius "r, since laF/BtlI < 1/2 there. To prove tp continuous continuoU8 at some lOme a' E E U, for any laF/1JJI1 consider the lame lI&Jlle problenl problem as • > 0 coDlider &8 in the 8tatement atatenlent of the theorem, with (G, b) replaced by (G', b'), where II' b' - tp(o'), tp(a'), and I/ replaced by ita restric(a, tion to the open 8ubset of Et given by

+

o.

e

({z, 1/) y) EEl: E Et: z E U, b'l (z, u, 11y E V, Iy 111 - b'l

el. < EI.

The procedure uaed used to obtain U, V, tp f/J gives us, analogously, U', U' , V', tp', the latter being a function fP': tp': V' U' ---+ V' 8uch that !(z, I(z, tp'{z» tp'{z» ,. - 0 for all sz E U'. (In the present context the prime' does not indicate differentiation.) But we are dealing here with the restriction of IJ to a smaller open IUbeet ~ than given originally, 10 subset of Et 80 that U' C U, V' C V, and 80 so that 11/ - b'l < et for all 11y E V'. The uniqueness property of tpfP implies that 111 tp'(:r;) ... == tp{z) ,,(x) for all :r; U' , 80 that Itp(z) - cp(a') tp'{z) ~ E U', tp(a') 1< 1< Ef for all zZ E U U'.' . tp{z) - ,,(a') tp(a') I1 < e~ whenever z is in some lOme open ball in R of center a'. G'. Thus I1tp(z) Hence tp is continuous at a'. Since a' was an arbitrary point of U, the function f{J tp is continuous. continuoU8.

IT6 116

VIR. 8tJCCZ88IVIl 8trCCU8lVII APPROXlMA110N8 APPROXlIlA110N8 VIII.

Corollary (lnHrseJunetion (Inverse Junction theorem). real-fJalued Junction lunction theorem), l..et Let,g be a real-wlued on an open. BUbBet 01 R Uw.t that contains the point b mad and 1UPfI086 suppose Uw.t that rI g' ezim eziBtt and IUbset, with ,'(b) #tMre e:Nt is continuoua contintulUs on thia thiB open subset, ~ O. Then there exist open intervals va18 U, V in R, with bE V, lUCk BUCh Uw.t that ,g is iB defined at 6IJCh eadl point 01 V and the reatriction 01 ,g to V is aa one-one mtlp function restriction map 01 V onto U tDhoae tDhose inverN inverse lunction ,,-1: U -+ _ V is di$erentitJble. g-l: differentiable.

open

l!JI consisting of all (z, On the open subset of Et (x, II) y) E l!JI Et such that IIy is which g is defined defined..we we define a function 1 by in the open subset of R on which, I(x, y) ... l(z,lI) - zx -- g(y). g(II). Set a == =: g(b). We may apply the theorem to this 1 and UI, VI C R, with a E the point (a, b) to obtain open intervals U., EU UI1 and bE VI, and a unique and continuous funetion lIuch that function '1': tp: UI - VI such g(tp(x» for all zx E U tp is one-one onfH)ne from U II onto tp(Ul) zx - g(rp(z» UI.I. The map 'I' rp(Ua) ... ('\ V•• g-I(U.) ("\ ,,-·(U.) V •• By the first proposition of Chapter IV the set ,-I(Ul) ,,-·(Ua) is an open subset lIubeet of the set on which, which g is ill defined, hence an open IIUbset subset of R. rp(Ua) ("\ VI is ill an open subset of R. tp(U rp(UII) is also Therefore tp(U g-1(UlI) ('\ l) ... ,,-I(U co}lnected, since it is a continuous image of a connected set. A. co,nnected, Aa a nonempty IIUbtlet of an open interval in R, rp(UI) tp(Ul) ill is itself an open connected open subset rp(UI»)' If we is the open interval (g.l.b. rp(Ua),l.u.b. tp(Ul), l.u.b. tp(Ul»)' interval (in fact it ill set U - U I, is a onfH)ne rp(U.), one-one map l), then the restriction of g to V ill l , V - tp(U tp and the whole of the corollary ill is proved onto U whose inverse map is 'I' tp. [It is only fair to remark that there are except for the differentiability of of'll. much more elementary proofs. For example, "g' maintaina maintains the same sign containll b, so that we can assume on some open interval in R that contains 888ume that that,g is either strictly 888Ume that IItrictly increasing or strictly decreasing. We can also assume ,g is bounded. Using the intermediate value theorem we deduce that g is one-one from any open subinterval of the open set on which it is defined onfH)ne inverae function onto an open interval in R. This enables us to define the inverse ,,-. continuous.] To prove prove'll g-l and to prove that g-I is continuous.) tp differentiable we may so that g'(II) ,'(y) .,A y E V; suppose V chosen 110 pili 0 if II Vi indeed this is true for the V we have constructed, conlltructed, and we could in any case guarantee this by replacing U and V by suitable open subintervals. Then for z, x, XI Xl E U, X x '" .,A Zl, Xl, we have

U.-

,,-1

X X-

ZI = g(rp(z» )g'(8), X• ... g(tp(x» - g(rp(xa» g(tp(x.» = == (rp(x) (tp(x) - rp(za) tp(x.»g'(8),

some 89 between tp(x) rp(x) and tp(XI). rp(ZI). Since 98 E V we have ,'(8) rI(8) ,,0 for 80me .,A 0 and we may write tp(x) rp(z) - tp(XI) rp(ZI) X x -Xl - Zl

1 ... =

g'(8) . g'(9)

....

Since'll Since tp is continuous we have lim 89 = == rp(x.), tp(XI), 110 80 since g' " is continuous we z.+"., have lim g'(8) g'(9) = == ,'(rp(XI». g'(tp(x.». Hence

....z. ·"·1

rp(z) - tp(Xl)... rp(z.) lim tp(x) 1 zX -- ZI Xl g'(f(J(X.» •

_., "'''.

13. ThU8 tp Thus"

DIPnJUllftUL IIQVA'I'lOJq IIQVA-nONa DlrnDll'nAL

177 In

is ditTerentiable as W&II we. to be abown. shown. differentiable at each %1 ~I E U, ..

In the COUl'8e coune of the above proof the equation

- ,,(~z» "(~~»

,,'(z) ,,'(~) -

was W88 obtained. This Thia equation can itself iteelf be coDlidered an immediate 00DIeeoroUary, since once it is known thU __tiable, quence of the corollary, that " ill is diI diflereDtiable, the application of the chain rule to the equation s~ - ,(.,{~» ,(.,(s» liv. 1 - I'(,,(2:»,,'(z). "(fP(%»~/(%). The above implicit function theorem and invene inverae function theorem generalize to function. poeralilatao. functions of more than one variable.. Their pnenlilatioDa win nece88&l'1 pre)iminariel preliminari. on wiD be proved in the next chapter, after the neceaarr partial differentiation.

IIS. I. EXISTENCE AND UNIQIJENUI UNIQUENUS TIU.ORBM8 TIDORBM8 FOR ORDINARY DIFI'ERINTIAL DDTEIlINTIAL EQUATIONS. Suppose that J is a continuous.reaI-valued certaiD open OpeD eontinuoUl.real-valued function on &• .-taiD subset .we the aubset of Et ~ and let (0, (G, b) be a point of this open 1Ubeet. IIUheet. To 101ft differential equation dy

d" -/(z, (h 1/) .... -J(~,fI) with the initial condition 1/(0) differentiable nal-valuecl reaI-yalued Y(G) -=r: b means to find a dilerentiable function "tp on some containing G a IUOh such tbM that for all •:I ill m lOme open interval in R eontaininc this interval we have f>'(x) ,,'(~) - fez, /(., .,{s» thi. ,,(s» (this (thil impIiea impliea tbM tha' the poID' point, ,,(~» mU8t must lie in the liven open subset (z, fP(z» subeet of B') lP) and in Idditioa MlditioR .,(.) .-<.) -- ••t. We note lint lO1u~oll " can be de8ned deftned fimt. that the interval in R on which a eoIution amall, even if the function / ill may be rather small, is defined on the "hoIe whole of .. • and is very nicely behaved. For example, for lor any lIOlutioD solution "tp of the dilerdifferential equation entialequation

we have

tan-II ,,(.) d t&nft(s) tis -

1 (f'(Z»1 1 + (f'{i»'

,

tp (~) - 1, .,'(.)

that tan-I .,(z) constant on any opeD open interval on which "fP ill is ,,(~) - % ~ is eonatant defined; if we impose impoee the initial condition .,(O) __ .,(0) - 0, then the only 0D1)' lOIution eootaining 0 is given by .,(z) .,(~) - tan •• on an open interval in R containing s, t.b\1l t.hUl reetricf,. nwtrictUI to I:,; I~II < 11/2. ing us r /2. Therefore if we are interested in IOIviOl I01viDl the above

80 80

o.

171 I'll

ftIL ~ DPIIOZIIIA'I'Ion APnODIIATlO.8

cWr. pneral diI. .tial equation with initial condition, all we can hope for in paet'Il II that lOlution exist IOtJN open interval containiq con~ (J. (I. Thie This is indeed .. tba& •alOlutioD exIat. on .... the CM8, Ihan not pIOV8 prove thie thiI f-' fae~ CMe, with DO no further eonditioDl, conditicme, altho. we iban ill lO1utioD to be unique, which II iI highly hiPly in tbiI tIaiI ted. tat. However if we want the lOlution deIirabl.in maD7 ._ IODle further funt. conditioDl oonditioDl are nee.Fry. For eiample, deIlrabi. in IDUl7 . .,. lOme are~. _mple. 11I and if 1(., I(a, ,) - 81,1 lO1utioDl ... 81 ,ltII aocl (., b) - (0,0), we have the two lOlutiooi 0 aacI Ma) - ::t. z:I. Bence IOID8 condition mUit muat be impaled impoeed on lif III we wiIh to aocl ".c..) .....tee puaotee •a unique 101ution. lOlutioD. The condition illlpCJWOD I it foBowilll eo-caIled LipdiU Lipd_ conditioD we ahaU abIIlllDpCIII'on ia the followiqllO-Cllled concIiUon: M E B such that. that wbeaeYer whenever (II,.,) (z, ,) and (II,.) (s,.) are in ~: there then exilt.I exiata ME. the open 8UhIet ga on which I/ iI ia cIe6necl defined we have bave IUhIet of ..

"'<<-)-> -

I/(s, JJ) -/(s,.) 1/(11,.,) - I(s,,) I SAIl, ~ MI, - .,. '1· This 8//'" exists Thia condition oonditioD iI ia automatically .Wed _tiafied if 1//" uiet. and ia bounded in the liven ga and if a vertical line 1IfIIID8Dt. Ieplent U. 6. entin1y entirely Pv. open opeD IUbiet sublet of .. within thie this cue c.- the mean m-.n tbia open .t let whenever ita it. extremitiee extremitiel do, for in tbia value theorem enabl. U8 \II to write (v -.) I(s, ,) - I(s,.) - (,

-I-:t

(s, ,,), II),

for IOID8 IOID8 ,, bdw-.a bet.... , a.ncI _ • (if (If ," - "I, we take ", - ., " - .), 10 10 M may be t&ktD to be any upper 1Jound IBlI"I. tak. bound for 1811"'. Theorem. be G fvrt,diDr& on ,. opeR . _._ 0/ 7'laeGrem. Let Lee J I_ a contiAuoua CORM.... real-NlU«l ~ /wf,t:liIw& - OJNI' . 0/ ". IAtJl 8u",.. "..,. tMre . E •R . . IAae IAtJl .. IAae conIaiftl contaiu 1M poiftt poaftt (a, b). 8"",. . .. " M. E ltd

I/(s, ,) -/(s,.)1 . I/(z,,) - /(z,.) I ~ MI, .Ill, -.1 --I

........ <-, ,) ..

are ita 1M ,.,. . . . . . . (s, ,) _ (s,.) (s, .) aN "" . . . .",. opeR ... ... "AM TAM "..,. lAIr. ..." mae. AIt E ., R, •A> 0, . IAae "..,. . " .... onlJ .... . IAtJl tINr. . aUea one -_ ",.. OM reakaluecl r ~ /tIrtI:liIIA /tmt:titm "fI on OR ,,'(s) - f(., (a - A, •a A) . . . , tp'(z) J(z, ,,(11» ~s» on lAit lAiI ..,."",., iftImHJl _ ,,(a) .,(a) - b. •.

+.) .- ..,

continuoua real-valued funcUon function" For a coDtinUOUl " on an open interval in •R that tba, contaiDi a, the equatioDl equatiODI ,,'(s) -/(s, ,,(s» contalDi fP'(s) -/(s, ,,(s» and ,,(a) .,(ca) - b hold bold if and onI7 only ,,(s) I(e, .,(1»)41 "(O)cIC b, u fo11owll followa flOm if .,(s) /(1, from the fundamental theorem theonm of calcuIua. ThUllOlvilll ThUllOlvina the liven dilenntial ealcuIuL cWrerential equation with initial condition .. eqaivat.t to IOIvioI IOlvina the "in~ Hln~ equation" II equivalct

J:r.

+

,,(s) .,(s)

r.J: I(e,

1(1, "(O)cIC ,,(I»)cII

+ b.

II to a funct.ion function # #I we ""te . If UIOCiate another function F(",) F(~) whoee whose value at any sz 1(', #(0)'" b, we 1188 lee that IIOIviol eoIviol the integral aqua.. (I'(#»(s) /(1, #(l»tlI equa-

f:

+

,3. 13.

DlrnUNTlAL ~UATIONII DIFnUN'lUL _QUAftON8

tion is the GOD

179 119

F(tp) .. fP, 11', that is, solving same 88 finding a function II' fJ such that F(f(J) a kind of fixed point problem; this is the basic idea of the proof, which we lOme N E H now proceed to work out in detail. We begin by ,choosing some H, r > 0, such that the open ball ·, such that N > I/(a, IJ(4, b) I, then some r E R, in ga E' of center (a, b) and radius r i. is entirely contained in the open set on is defined and such that 1/(2:, I/(z, 1/) II 0, 80 A < r/2N, and AM choose 1& E E H, 1& so that It, h < r/2, h hM < 1. The rectangle all

{(z,1/)EE': Hz,1/) E.EI : Iz-aISh, Iz - al S h, lu-blSNhl 111 - bl S Nhl

I is defined and for each is then entirely contained in the open set on which 1 (~, 7/) in this rectangle we have 1/(%, tI) 1< (z,1/) lI(z, 1/) I < N. We are going to prove that there exists one and only one continuous function II' f{J on the closed interval (a - 1&, such that [0 A, a + h) 8uch tp(z) ... ,,(%)

f:f I(t, /(t, tp(t))dt ,,(t»dt + b

+

for all 2: A]. To do this, consider zE E (0 - h, ca + hI. collSider the complete metric space C«(a 11., (Ia h]) functions on the compact C«(c - A, Al) of all continuous real-valued fUDctions hI, 88 at the end of. of Chapter IV. Let B be the metric space apace (a (0 -- h, a 0 + h), closed ball in C«a C«(a - h, G Ill) NA whose center is the constant 0 h» of radius Nh function II, b, that is B is the set of all continuous functions funotion

+

+

+

1/1: (a - h, atJ "':

h]-+(b + A]-+ (b -

Nh, NA, b

+ Nh]. Nil].

subset of a complete metric space, B is itself itaelf a complete Since B is a closed 8ubset metric space. apace. We claim that any solution of the above integral equation must lie in B, that in fact if II' fJ is ... u above then Itp(z) l.p(z) - bl < Nh for all E (0 (a - la, lal. For if there exist points pointe :,; z E (0 (a - h, A, 0a Al. A, 0a h1 A] such that sz E Itp(s) NA, let 'Y I,,(z) - bl ~ Nil., "'I be the greatest lower bound of Is Iz - 01 01 for allluch anluch pointe. points. Since "fJ is continuous and "(0) tp(a) .. - b, it follows that 'Y "'I > 0 and Itp(a:t: 'Y) - bl- Nil. Nh for at leut leaat one choice of the siln :i:. Thus Nh I,,(a :t: "'I) sign :t:. I'tp(a ,,(a :t: ,.) "'I) - 11'(0) "'III" (ex) I, ex between G a and a:i: a:t: "'I, ~(G) I-I ""'(Q) I, for lOme IOn18 a 'Y, by the mean value theorem, and the latter expression equals I'Y/(a., 'Y/(ex, tp(ex» I < "'IN S AN, which is a contradiction. Thus any solution II' 'YN I(J of the integral equation is in B. Now for any ~ E B define a new function any",

+

+

1-'

,,(a»

F(",): (a F(1/!): (0 - h, A, a

-+ H + h)A]-+ R

by

(F(t/t) )(%) (F(I/I»(z)

J.a /(t. t/t(t»dt f I(t, ",(O)dt + b.

Since 1/1 E B, for any t E [a - 11., A, a0 + h] ~ Nh, 80 that 11.] we have 11/I(t) II/I(t) - bl S Nil., so ... a fUllction fUllctioll of t, and I/(t, 11(t, 1/1(0) !/I{t» 1< I < N. I(t, ",(t» "'(0) is defined, is continuous as

1. lID

Vln. 817COM81ft VIII. 1I1700J1M1 . . APPBOXlIIA'ftON8 APPIIOXIJIIA'l'I01f8

+ 1&], A], (F(~»(s) (F(~»(s) is defined and I (F(t»(z) bt(F(~»(s) - blII LBince F(~) F(tJ-) is clearly continuous we £-/('I(t,, tJ-(t))tltl ~(e»dtl ~ Nlz Nis - 01 til ~ IaN. Since

Hence for A, (J lor zs E [0 --1&, tI

have F(t/I) fA) E B then for any element F(~) E B. Thus F: B .... - B. If '/I, ~, wEB A, atI + A] II] we have z:e E (a (0 - 1&,

+

II(F(~»(:e) (F(#) )(s) -

(P(faJ) (F(w»)(2:) (:e) I

-I J:f: (J(', C/(e, tJ-{e» ~(t)) - i<" i(i, w(t)))dt w(e»)tIt I

:s; ~(t» -Ie"~ - !(', wet)) CIJ('» I : •'E All :S I- - -I til max 11/(', (11<', ~(t)) e [a[s - A, atJ + An ~ IIz -- alMmu a IM max ((I~(t) 1~(t) - w(OI w(') I : te , E [tl [0 - A.a A, G +1&11 + A] J

S AMd(~,w), ~ AM tI(~. w), where dtI denotes the metric in B. Thus

d(F(tI'), 1& M tI(~, d('/I, ",). tI(F(~), F(w» F(w» S ~A w). But AM

< 1, 80 110 that F is a contraction map. The fixed point theorem thus

&MUreI fJ E E 8I8U1'8I U8 us of the existence of a unique unique"

BUell IlUcIl that ,,(:e) fI'(z)

i8 B 8uch IlUch that" - F(,,), that is

£-/(' L-/(',, ,,(O)dt fI'(e»tIt + +"b

+

fo'r all aU %_ E (a [a - A, atI + AJ. for Al. For :e z E (a - A, a0 + A) we clearly have ,,'(_) -/(s, -/(s, ,,(s» - b. Thus the existence part of the proof is ,,'(s) f'(s» and ,,(tI) "(0) .. complete. However it is not Dot immediately obvious obvioUl that the reetriction feltrietioD of fJ to (a - A, G + A) ie eolution on (G (0 - A, a G + Ia) 01 our diI_tial diflerential "to is the only on1ylOlution A) of concJition. To Me equation with initial conclition. lee thia, this, note that the above proof would lODe throUSh tbrouP with IIA npIaoed have lOne replaced by any A. Al E R llUoh luoh that 0 < Aa Al < A. ADy 101utiOD eolution on (0 (G -- A, a8 + II) A) of the diI_tial AD:1 diflerential equation with initial ooncIition liv• PVII• a lO1ution eolution of the intepal intesral equation on [a ooDClition (8 - ha, "'1, G + Ai). All. But intesral equation hal a unique lOlution on [a - A., we know that the intepal At, GCJ + AiJ. AI). eolutiODl on (8 (a - A, a8 + A) bave Thus any two aolUtiODl have equal reetriotioDi reetnctioDl to [a - AI, a + All. AaJ. Since this thie is true for all aU A. IlUch that 0 < A. [ea At such Al < A, la, there is moat one IOlution eolution on (a - A, II a + A) and our proof is DOW at most now complete.

+

+

+

+

+

+

+

The preceding theorem can be generalised to systems 8ystems 01 of fi!'lt fint order 01 the fonn differential equations of

'I, ·... · .,, ,.) :~ -/l(z, -I.(s, '"

:

-/.(s, '" ···,,.) ... , ,.) -I.(z, 'I,

":; l/., ..• l/.) ~ -/.(s, -I.(z, 1/1, • • .,, ,.) conditionsl/.(a) -~, ..... with initial conditions 1/.(0) ., "., i-I, i =- 1, 2, ... , n. ft. Here functiona/ functions /1,•.! II, .•. , variablee are liven, pven, toptber &a, ••• I. of ft" + 1 variables topther with reU real numben Dumben ., tI,", ...,, 6.,

t a.

DUnUIl'IWt 8QV.'IIlOIIa

ID

Ex.

and the problem is to find functione rt, ..• functiona ft, ' , , ,,. ,'" of •s atWJiq l&tiIfyiq the &iftD pno equationl. equatiooa, Except for not&tioDal notatioDai complicatioDl, eomplicatiooa, h\he pDeI'IIiIatioo pDII'IIbatioil of 01 the aIao in. inter.W preceding theorem is straightforward. However we are a1Io . . in getting neultl than P'ting abarper ebarper results t.han have 80 far been obtained fw for •ft -- 1, 10 we beIin begin with a rather specific lpecific lemma that iIolatel iIolatee the h teclmiall technical detaiJI detailI nIatiDc N1atina to the fixed point theorem. £em..... Ia, .•. ,1. be Lem..... 1M La II, " "I. ". ..................... conhftUOUl ............, /vNJfiIIu /vIMJI;iIIu .. - .,. .,. ..... ••• , "). '-)• .&qIpoM " ,. . .tAft . . ...... ... ..., U of .... B-+I C1acd Uaat cont.aiu conItJina , 1M. poilU (0,'" (a, &., "" N, MER ItICA Uaatlor C1acd lor . . i-I, N,MER''-1, ... ,. ,ft

1/,c., ,., 1I,(s, Ul, " ", ,.>1 ",)I < N ~ (."., (s, Ul, ... ' , "".) U.) E U ancf aracI ......

«(,. - .a)1 ,., + {f.
1I,(z, '" Ul, ...,,.) , , " 11.) - l,ca, I'(z, -., '1, •.•,..) ' , " ..) 1ISM 1I,(a, ~ M(
u,

....... IAaC ~ (a, (s, ,., ft, ••• " ", ,.), U.), (a, (s, .., II, ••• ' , ", ..) .. ) E U. 1M La AE AE R, A> 0, IN ". ... auo\ lUI

,a, ...,,.)

(a, fez, '1, , .. , ,.) E"" E 8'*1:~ I. Is -

01 ~A, SA,

I,. ••• , I,. 1'1 -- btl ~ S NA, "" I", - "-I b.1 ~ S NAI C

U.

TA4m if Vi < 1 "..,.. " OM . .pIe (fIl, (....... ,,,.) of T1Nft V AM v'ft IIun .... . . . one ancf aracI . cmlr one .....,. •• ".,.) IAa& I.. I.. . . ,A, a + A) . . IAaC tid i 1, ... ., .,,., ft, tp4'(a) ",'(z) --/'(s, l,ca, .. fPl(s), (a), ••• ' , .,, ".(.» ".(s» •em 'fAit '1Ma .,."., iftIerNl ancf aracI ..,{a) tp4(a) - ...

~ Oft 1M tIae ifttenJGZ Co ~ /uftt:li.tma ~ (a

w. want to find functio....... ,,,. atiafyiq eatilfJiq the , . . of We functiona fPl, """. h . I)'Item 01 in.... in..,.. equatione equatioDl fP4(.) ",(s)

J:r. l,c" l,e" ..(e), ••• , ".(e»)dt +." +~ fPl(C), .. "".(,»)dt

i-I, •..," ,1, ... , '"

AnaloaouaIy to what "u of the pnoedina theoraD, we AnalOlOUl1y WII done in the proof 01 ~ theora, eoneider the h comJIM' eom~ metric apace [a to - A, a + Al A) and the h oomplete complete . met.rio CODIider ... apace IF of all continU0U8 eontinU0U8 functions functiona from [a (a - A, a + Al A) into .., B-, IUI M the We indicate a function into .B". by it. its.tuple oomend of Chapter IV. W. .tuple of .... functiona, 10 that an element # ~ 01 of If a-tuple (#t, (~ ••• , .J, ~.), ponent functions, IF is an tHuple ~, is a eontinuOUI where each #, continUOUl real-valued function on (a - A, a + A) AJ aad ADd la, a + ~s) - (~(.), (~(s), ••• . , • ,I ~.(s», CoaaIder CoIIIkW the tile for any as E (a la - A, + Ia) Al we have #(a) au.... B of IF of all # eubeet If COJIIiet.inc coneiatinlof ~ - (#., (~., ... , , ", #.) ~.) IUCh eucb that I.~.) t#~s) -"1 -'" ~ S 1IA for all .Ela sEta -",a+AI -A,a+A) and all .-1, '-1, ... .. "ft, ,A. In the~. . . . B u it WII wu in the pnceding precedilll proof, but it . is not a ball in If, .. '- •. . a ..... aublet eubeet of If: for if '/I',.p,.p, #I, #I, #I, •.. '" is a sequence eequence of element. elements of B that GODversverpe to the element # of IF If then for each •z E [a (0 - A, a + AI A) we have lim #'(a) t/I'(z) - #(a), ~S),

#.(.».

,.. ~

10

t.hM

,..

lim #i,ca) f',(z) -- #,c.), ~,(S), i-I, i - I "... " ,, ft, ~

+

_

9IIL

~

&JlnOJaIIAftOIiB

and - bit S S Nil Nit for aU j we can Ultnfore therefore aDd from fJom the tile inequaliti. iDequaliti. 1#1.(%) I~'(*) -'" - "" S NA, 10 that fEB, '" E B, thUl thus verifying the criterion for deduce I#.(s) If.(*) -'" c10mre III. Since B is a closed aubeet subset of cbure of the theorem of '3 ,3 of Chapter IlL •a complete metric epace, space. We claim tbat that apace, B is itlelf iteelf •a complete metric apace. (fII, ••• , ...) ".) E IF ayatem of integral equatiOD8 if ,,fP - (A, tF .tides the above .)'Item equations then aD *~ E [0 (0 - II. h, 0G + h) II] and tbID "E B, and aDd in fact IIfP'(z) ~(*) - bil '" < Nil NA for aU aU ft. For if there exi8& exia' points (ea -- A. la, 0a + h] A] auch such that all .i - 1, ••• ,,tL pointe ~ E [0 I.,.{z) ~I ~ NA for 101118 lOme i, let -r -, be the pateat ~test lower bound of 1* Iz - al I~*) - 6.1 lor #PI is continUous continuous and ",(0) ~ we have for aD all such IUCh point. pointe z. Bince Since each ~ ~a) --= .. -r > 0 and lOme ii-I, •.• ,," I. ., one aDd l.-c(a:l: '~(a. ,,) -r) - 11.16.1- NA for 101118 - 1, ••• ft and at leut 01 NA -, :I: -r) 'r) - ~-) ~G) I-i-rw(a) I-, 'rfJl(a) II,, of the two ahoieee chol_ of lip ::t:. •. ThUi ThUl Nh -I .,,(a ~- • IOIDe cr CI between bet. . . G _ and a . -r, by the mean value theorem, and the for lOme G2: latter exp~ equa1a I~,(a, ".(a), < -rN S IaN, AN, a contralat.ter exprlllioD equaIa h1,(a, fII(a), ... ••• , .,.(a» ".(a» I <.,N diction. equations on [_ (a - A, '11wI any eolution aoIutioD "tp of the ByBtem Bylltem 01 of integral equatiooa dietion. ThUl ia in B. Now for any fEB A) ill # E B de6ne define another ft-tuple A-tuple of functions r(#) - ('s(#), ,.(~» from [0 (a - h, A, a G 1'(#) (I'.(f), ••• ... , I'.(f» A] into R by

_tid.

+

*

*.

•_+

+

(I''<~) )(s) - /.. 14(t, f.(O, ~I(t), ~ .. , f.(e»dt ~.(t»)dt + +"'. (I',(f»(*) /.-/.(" It.. 0• • • ,

8inoe ~ #E for any t E [a [0 - A, a + A) Since E B, lor A] and any i-I, i .. 1, ••• ... , ft A we have I~,(t) 10 that Ie', f.(O, ~l('), ... ••• , f.(t» ~.(,» is ie defined, is continuoua .. f,(O -- ~I '" S NA, 10 ..& function 1/,(1, f.(O, ~l('), ... ••• , f.(e» #.(1» I < N. Bence Hence for * z E [a (0 - A, la, functioD 01 of e,I, and 1/,(', _+ Al, AI, (I'.(f»(s) G (F,(~) )(s) is ia defined and

I/..

I

1 "'I "" /.(t, fa(O, ~I(t), ... ••• , #I.(t»dt I (1''<#1) (I',(f) )(s) )(*) --'" ~ 1/.- /.0, f.(e»dt I S N1&. NA. IMh ,,(#) I'.(f) is ie clearly continuous, we have Since each bave I'(f) F(#) E e B. That II, 1': B .... B. We now f, wEB. F: S DOW ahow that F is a contraction map. Let Let~, '" E B. For ~ (0 -- A, G + It] - I , ... , ft, aD7 •s E l0 + A) and ii-I, (I'.(f»(s) - ('C1I I ('C<#»(z) (J.(t,, fa(O, (J.<' ~I(t), •.• ••• ,, #.(t» #1.(1)) -/.(', - I.<.t, ,..(1), Wa(t), ••• • • .,, w,,(t»)dt ~ Is -I DIU 11/.(1, fa(t), ... , f.(t» S I. calmu II/~" #1(1), • - -, (0, .• E [0 --1, A, a + All -/'<', ... c.Ja('), • ••", ...(t» (,» I : ,'E A) J ~ la 1* - calM _1M max DIU Id(t(i), I"(f(i), tt(t» E (0 :s; flJ(t» : t'E (a -A, - A, 0G + + AU All

-I J:' -I/.·

iI.('»

,,,,,(0)>.1I

AM w), AJI "(#, tI(~, .). UI8d the ame (We have here UIed eame letter" letter II to denote the metrica metries in B" E- and in B.) ~

'l1luI f t.

"«I'(#»(s), (I'(.,»{s» (I'(w»(*» - ( "«I'(+»(s),

t. t.... «I'.
Mv'ft "(f, tl(.", w), S 1&AMVn 10

"(F(f), F(w» SAM tl(F(~), F(II)) S 1& M Vi Vii "(",, d("" w).

(F,(w»(s»lr (Fi(W»(S»lr

'I.

Dlft'DIDI'IlI.L JlQUAft0lf8

.

18

Since we have UlUJned AM Vi < 1, 1J ., , is indeed a contraction map. 888UJDed that 1&Jf Thus by the fixed point theorem there is a unique" unique f(J .. :lID ("', (\Oi, ... ••• ,.,.) , ".) E B such that "fJ ..' auch that - F(ffJ) F(,,),J that is, such "",(z) fPi(z)

~

+

I.{', ",(0, ... , .,.(t»dt /.- !.{I. fI'1(O••••• (/Js(l))dt + b, b.

for all sz E [0 h) and all ii -==I1, ft. For z E (1.1 (a - h,o h, a + h) we (0 -la, a 0 + Ia) , ... ,J n. == !,(z, fJ.(z» and "",(1.1) ~(o) ... == bb"i , so the existence clearly have f)l(z) "",'(z) ... I,(z, f)l(Z), fI1(z), ..• •.• , .,.(z» part of the proof is finished. restrictions of "', 'Pl, .•. ••. , ". fP. &nished. To prove that the reetrictions to (ca + 1&) are the only functions with the desired properties, note (0 - A, G 1.1 +!&) that the above proof woUld have gone through with Ia It, replaced by any Al Al < A. (0 - h, A, II G + h) of the system A. E R such that 0 < Ia. h. Any solution on (1.1 of differential difterential equations with initial conditions gives a solution of the system of inteKn1 hi, 1.G1 Ad. AI). But we know that the system of intepal equations on [0 (1.1 -- h., intepsl hi, 1.1 0 + hi). hl1. Thus any two intepal equatiODl equations baa a unique IOlution lOlution on (0 (1.1 - h., eolutioDl A, 1.G1 + h) ·of 8ystem of ditTerentiai differential equations with lOlutioDa on (0 (1.1 - h, of the system (0 - h., 11,1, 0a hi]. Since this is initial conditions have equal restrictions reetrictions to (1.1 hi). tNe most one solution on OD (1.1 (0 - h, true for all Al A. sUch that 00 < h. h, < II,h there is at mOlt CI A) II h) and our proof is complete.

+

+

+ +

+

Genera1ising our previous definition, we I&y say that a real-valued function Generalizing

I/ on an open subset sublet of E-+l satisfies aatia&es a U,*"ilz Lipschitz condition if there exists a Dumber ••• ,, Y.) and (z, (x, '" '1, ... , ,.) z,,) are number MER 8uch such that whenever (z, Yl, y., ... in the open set Bet on which IJ is defined we have

«,. -

Il(z, ,., IJ(z, til, ... • • .,, ,.) fl.) -/(z, - f(z, '" '1, ... • · .,, .. ,.)) 1 IS M M«1/1 - ,a)1 '1)1 + ...

+ (Y. (ti.

- .. )1)'/1. ",)1)1/1.

This condition oondition can be given in another way, for lince Thia since

1111 -1111

_I + ... + I,. If/. - .. ..II ~ (w. (U/l - .a)1 ,.).1 + ... · ·· + (,. (1/- - .. ..)1)'11 )1)111 (1,.t ~ max { Itil - '.1,···, '11, ··· hi. lII. -- .. .. II} ~ 1.(1,. - •• 1+ ... + I,. --z-I), .. I), ~l(lfll-'ll+ +111n J

eatiafies &a LipschitJI we see that I/ eatisfies Lipschitl condition if and only if there exists such that whenever Ifie is defined at (z, Y" M' E R luch 1/1, ••. ••• ,J Y.) U.) and (z, (x, '1, ZI, ... ••• ,, .. z.)) we have

I/(z,Ill, ... ".) -/(Z,'I, ... , ..) 1S M'(IYI - 'II +

'" + I". - .. I).

A.. a conaequence oonaequence it is possible to state Aa 8tate that a rather large cl&88 class of functioD8 functions tatiBfy Lipschitz LipschitJI conditions: conditioD8: a real-valued function If on an open subset of eatiafy 11'*1 satisfies I&tisfies a Lipschitz LipschitJI condition if al/aYl, B-+I a/laYl, ... , af/ay. a/lay.. exist and are bounded on the open set and if whenever (x, lIl, Yl, •.. ..• , 'II.) lI,,) and (x, '1, ZI, :: .. •• ,, z..) are in the open set so 80 is the entire line segment seglneut between these two points.

184

VUJ. SUCCES8IVIJ SUCCBSIIIVII APPROXIMATIONS VUI.

For tI.. ) -/(x, - I(x, zh 21, ••. ..) I1 I/(x, til, 111, ..• • • •,, tI,,) • • • ,J 2 z,,) ~ 1/(:1:, I/(x, 1/1, til, ... tI..) - f(z, I(x, -I, 21, 1/1, til, •••• tI..) 1I S • • .,, 1/,,) • .,, tI,,) + If(:I:, If(x, '1, 21, 111, YI, •••• ,1/..) - f(x, 21,~, 2" 2t, 1/1, til, •••• , tI..) 1+ 1+ • .,1/.) • .,1/.) •••• ,, "'-1, ",,) tI ..) - f(z, f(x, -I, 21, ··., ••• , ••) ...) 1 + If(x, 21, ZI, ••

+ +

- 1(1/1 -I

21)..2L (x, 'II, "1, 'I, til, ... ••• ,, II,,) I1+ _1)..2L (z, iJyl 8111

+ I1(Y.. (Y.. -

.....·

.....·

z.)) ...!l21, ••• , "'-1, %a-I, ".) .".) II,, ... ..2L (x, ZI, iJy. By"

n, 'I. ", is between ", and "2, (or equal to them if these where for ii-I, .. 1, ... , ft, so that if M" Mil i8 is an upper bound for IlalJaYll, ... , Ilafla" 1 latter are equal), 80 a//a1/11, ..• al/By... 1 we have

I/(x,1/I, ..) -- I(x, 21, I/(z, til, ... · · .,,1/ tI,,) f(x, '1,

... + I"" ... 1). '11 + ·.. If/" - .. I)·

•••· ,,In) ... ) IS I ~ M"(I M"(lYI .. 111 - 211

Theorem. Let II, It, ... ,I.. ,I" be continuous real-valued !,unctiom lunctiom on an open subset of Era+t fl, ••• /. Eft+t that contains the point (a, b1l ,, ..• ••• ,, b..). b..). Suppose that ft, ... ,,f" Lipschitz conditions, that is there exists E H BUc1& satisfy Lipachitz uiatB M MER BUCA that I/;(x, 1/1, ·... .. ) - f,(x, 21, 1/,(x,1/1, · .,, Yy..) I.(x, '1,

.. ) I ~ M«1I1 M (b/I - -1)1 2ill ... ,,2'ra)

(Y.. + ... + (fl.

...)1)1" ".)1)111

... , 11. n whencr'er whenCller (x, Yl, tilt ... ..• , Y ..) and (x, '1, 21, ... ••• ,, ,,,) z..) are in tM for i ... cr 1, ..• y,,) the given open set. Then there exists II.hE H, h > 0, such that there ezi,ta exi.ts one and only E R, of real-valued functions function. (eI'l, (c,I>t, ••• , !P .. ) on (0 (a - h, (Ja + h) BUCA BUc1& that one n-tuple 0/ \0,.) for i = 1, ... , 11. n we hat.e hcult cp/(x) !p/(x) "" li(x, f,(x, CJ'I(X), !Pl(X), ••• , "!P..(x» (.1:» on thi. Jor tAil interval and II:

epi( (J) = = b,. !PiCa)

chOO8e some N E H To prove this choose R such that

> max ({I/t(a, 1/1(0, b1,l , ••• , b.) I/.... (a, bt, ..• , b.) btl) 1}, b.. ) I, ... , 1/ II, then lOme some ,r E R, H, ,r > 0, such that the open ball in .8"+1 ga+1 of center N

(0, (a, bJ, bt, ••. , b..) b.) and ndius radius r is entirely contained in the open set on which 11, fl' ... ••• ••• , II".. If,(x, 1/1, ..• ,, 1/.) Y..) I < N for each i "" are defined and such that I/.(z, tit, .•• == 1, ... , n tI" ... ..) is in the ball. If II,h E ft, H, hh > 0, any point (x, "1, til, whenever (x, 1/1, ••• , Y flfJ) ... , 1/..) E ga+tsuch ~ h, IIYl bll ItI.. -- b.1 b.. 1S ~ Nh . · .,1/,,) E-+18uch that Ix - al S fit - b · · .,, 11/. Nil, l ' ~ Nh, ... will automatically be in our open ball if hi + nNIh,I nNII&' < rl. Hence the theorem lemma if we take the U of the lemma to be our results directly from lronl the lenlma open ball, take the same It, E H, fl' ... , I.., f .. , a, bbllt , ••• , b.., b.. , N, M, and choose chooee hhER, h > 0, such that h < r/(1 r/(l + nNI)lll nNI)11t and hM Vn < 1.

vn

ft, ... ,f continuouB real-valued Junctions function. on an open Corollary 1. Let 11, ,1ft.. be continuous aubset oj E.. Eft+! ••.•, bll)' b.. ). Suppo.e that 11, It, •· ....,, f. I .. I'Ubset +I that contains 1M the point (a, bhl , ••• Suppose tluJl aatisfy Lipschitz conditions. condition•. Then i/S if S i3 is any open interval in R H that contaim contains BOtilJ!Y 1A:pschitz the point a there i8 (~, ••. is at most one n-tuple 0/ oj real-valued Junctions functions (c,I>t, ••• , .,,,) !P..) on 8 for each ii = 1, ... , n we hove fPl(x) == f,(z, c,I>t(x), f(JJ.(z), ... ••. , f),,(x» S BUch such tMt that lor have !pl(x) = f,(x, !P..(x» = bi. b,. on S and !p,(a) .p.{a) ==

13.

1. 111

DIJ'J'IIUJC'IUL ~AftO... '8QUAftO... DlI'nUII'IUL

For suppose that ('Pl, (~, ... , ".) ("'I, ••• , "'.) t/!.) are an two tHuples n-tupl. of 01 ....) and ("'It functions each of which satisfies the &iven given conditiODI. conditione. W. We mUit abow that =r 1/11, ••• , f/J" accomplished by a• .,.,. very limple IIIIUarprp." == .., 1/1,.. "'.. This can be acoompliabed ment &8 uniqueDell pari pan of the as follows: We begin by noting that by the uni~ theorem the subset of 8S given &iven by a •• ,

"1 - "'I, ... ,

(a E 8 S : ",(a) {a ~(a) - ",,(a), #.(a), i-I, .• " ,,) fit I

I

IJ

"'1, "', "'..

is open. By the continuity of 'Pl, t'l, •. " .... f/Ja,, ~l, •• J ~., tbileubeet UUa subeet it iI eloIed. c1cJIed. Bince 8S is connected this subset IUbeet muet itlell or the 8IDPt7 empty ... let. 8inee Sin. must be 8 S itself the subset includes the point a, we are forced to the ocmcIusloIl aoncIwdoa. that it. it must be 8 itself. I

a

•• ,

Let I/t,.. .....,!. reol-wJltMJd /vrfI:litma ~ ,. an open , I. be continuoua reakalued • ,.

Corollary 2. J.

a

sub8et U of Era+1 that contains ba, •• •••"J ba). b.). 8uppotl 8uPJlOll II&at IAat Ia, /1, SUb8et 01 E"+I contaim the point (a, ba, sati8fy aatiBlII l./ipschitz lApachitz conditions. condition8. lAt Let N 1,I,

••• ,

.•• , I.

aueA ".", tMe N. E R be aucA

I/,(z, fll, aa.,, II.) tI.) I S N. 1I.{s, III, ••• ~ N,

u.

lor , ....•~ n and all III, ... be .. .open ,. for all ii --=I1, aU (s, (%, 1/1, • •• ,J II.) E U. Ld La 8 C R IN interval containing the point IIa suck t~nt"val such that thai, a

•

(x, Yl, .. a, y,.) E E,,+l : z E S, IYi -

"'I S Nilz - ai, i-I,. ,-"ft) C U.

Then there exiltt unique lundiana functioM 'PI: H, ... , .... f/Ja:: 8 8 -.... R . ... IAat for lor 'Pl: 8S --+ R, - .., . each ii -==I1, f{Jl(~) IIZ/,(%, f{Jl(2:), "', • ".(s».S.., ".(z» _S.." ~.) t'4
.,

If there exist functions~, functioDl'Pl, .. ••.a,,,,. properties . tbIQ ... with the stMed stated propert.i_ . . t.be,r t,hq Also the equatioDa must be unique, by Corollary 1. A1Io ",(z) "'(2:) ... -

f 1.(2:, I.(z, fP1(z), """' ",,(z»)dz be, 'Pl(s), "', .... (s»dz + "',

s

E 8, i-I, i-I,." ...0'. ,"

19'i(z)-b.I~N,ls-al, 80 that for all sE8 we wUI imply ICPi(X)-biISNil~-GI, will have (x, ••• , .... tp.(z» by (z, f/Jl(~), "I(Z), "', (z» E (J, Q, where (J Q is the I8t aet defined b7

Q - (Z,lIl, ... ,11.) E E·+I : s E 8, III' -

"'I ~ N,ls - al,i -

1, ... ,,,,.

(Bee Figure 33, which illustrates the cue " - 1). One ocmeequenoe (See OODIequenoe of tbiI thiI cps, •••• ,., I. GUtiide is that ~l, Ip.... do not depend at all on the valu. values of II, "', . · .• !. outside the set Q. That is, if we consider a similar problem, with all the same I&lIl8 data as at present except that the valUeB values of 11, I., ... I. are altered on U - Q, then 88 .•• ,,I. the same functions "1, "I, ... , .... tp. will aolve solve both problems. But to &0 10 through throu&h with the proof we must take into account the behavior of 11, I. outlide /1, ••• ,,/. outakle &0 to the trouble of modifying I., .. I. out.side a way Q, 80 we 10 modifyinl/l, ...",I. outside Q in auoh lOch. a

• ,

_

WII. IIVClClIIIIIftl API'IIOXIII4'ftON8

Ilope N

.-

r -

--

,,(~) .,(z)

Ilope-N IIIope-N

I

•

8

"..,..88. ,.... .. 0 ,...,..aa.Tbe

1(., r) e .. : a: eS.lr - 'I~ NI_ -III. for &be _ O-f(.,r)e":.e8,lr-"~NI~-ClII die out

"• - 1 "~ CoroIIIry CaroIIIry 2 .. ladicat.ed buIioaW by 1hadiDa. 1IIadiq.

that Ala auxiliari., functions "': ~: B'81- R, thal our proof will wiD work. All awdliari_, we define functiolll i-I, ... ,ft, , ft, by

N,lz N,I~

,-be> N,lz --01 --al+be cal + lJe if ,,lJe > N,I~ cal if I, bel S ~ N,I~ Nilz - cal 01 I" - lJel -N,ls -al+be if ,,-lJe < -Nils -01. -N.J~-caJ+lJe <-N,I~-cal.

{ ",(s, ,) ,.,(~,,,) { "

,,-bo

TheM funct.ioDl •.. , TbeIe f1lDClioDl ".., ~., ••• , ". "" are continuous and for any ~z E 8 and "., •.• ,. bave (s, ••. , ".(z, ". E R we have (~, "..(s, "..(~, II.), .•. ""(~, II.» E Q. Setting Betting

".»

MS,,., -/rt,z, "..(z, ,.}, ... ••• , ""~, ".Cz, II.» f4(~,,,., ...,,.) ... , ,.) -/'(~, ,..(~, ".), B, ,., ra, ..••0 0,, ,,.. E R and i-I, oO't, ft, we pt for Ie s E 8, get each "g, continuous, ,,.(S,,,., ...., fa) r.) 1 JS N't aod and ,,(s, ,.(~,,,., -/'(s,,,., 1"(-,,., ~N ra, ••• ,, ".) fa) -/,(s, "., ... , ".) If.) whenever " let MER (_,,., ... ••• ,, ,.) ,.) eo. E Q. Now lUeh that (S,,,., II e R be INCh 0 0,

1/'(-,,., If.) -/,(z, '., ••• , .. ..)) 1 SM .}I + ... + (fl. (,. -- .. )1)1/1 1/,(.,,,., ..., ... , II.) -/'(~, '" I :S II (W. ({JI. - •.)1 ..)1) III ••• ,,,.), (s"., (s, ••; ... , ..) ..) e for all aU (s,,,., (S,,., ...".), E U and i-I, ... ,ft. It ia is •a fact ,,), (z,.) (~,.) E 81 B' and each ii-I, - 1, •.. that for each (s, If), ... , ft we have bave

I".,(s, ,) - ".,(s,.)1 s I,III -., - al ; I,.,(s,,,) ,.,(~,.)/:S to prove this it aufticee IUppoee fI ~ a, ",(z, ,.,(:r, .), IUflices to euppoee "~', ",(~, II) ,. ",(z, ,~~(s,,,) ~ "'(~, ,) ~ ~ ~s,.) "'(~, .) ~ a.

..

"

80

that tbat

ThUl for any s~ E 8,,, ..... la, ••• , .. E Rand i-I, 8, "., ... , II., fI", ••, i = 1, ... , ft, we have I"(s, ra, ... ,,..) I/a) -- ,rt,z, g,(~, •• .0 ."On . IMS,,., a.,, ... ••. , ..)>/I -1/,(:r, fl.>, ..., -I,(z, ,..(z, 1'1(Z, .tl, ••• , ""(~, JAa(Z, .. >11 -1/'(s, "..(z, ,..(s, II.), .•• , JAa(Z, ""(~, fl.» -I,(~, • .), ... ':')11 :S M«,..(~,II.) "..(s, .tl)1 • .»1 + ... + (p,.(z, (,..(~, fI,,) ,..(~, .. »1)1/1 S M«Pl(Z, 'tl - "..(Z, !I,,) - ",,(Z, ...»1)111 :S •.)1 + ... + (". (II. - ..)1)111, ~ M(
".»

•'3. 3.

Dl....ZUH11AL ICQUATlON8 ItQUATlON8 DlrnUN'ftAL

187 18'7

'1, ... ,,.

80 (~onditions, with the same sanle M. }'f. Since so that Ill • .•. , II. &l80 also satisfy Lipschitz conditions.

Ii may, if necessary, replace each fi and gi Ii have the same ~tl;ctions restrictions to Q we may. Ii stateolent of fi by gi IIi so as to be able to assulue 888ume that the open set U in the statement the corollary is I(x. (x, Yl, y,.) E SI. 81. Making l\'laking this 888umption. assulllption, YI, •.. ••••, Y ..) E E,,+l : x E

+

choose h E H, h > 0, such that S oontains ~losed interval [a - h, a + hI hER, rontains the dosed and such that h M < 1, M being the constant of the Lipschitz condit.hat given any open subinterval of S of tions. Then the lenuna lemma tells us that length 21& S there exists a unique solution of the 2h whose WhOfie extrenlities extremities are in S system of differential equatioll8 y/ ... == !,(x, ... , y,,), i == = 1, ...• ... , n on the equations Y;' f,(x. Yl, ••• subinterval with arbitrarily prescribed values of Yl, ••• ,1/.. Yl••••• y. at the center (~I, •••• ••• ,~,,) of the 8ubinterval. subinterval. Suppose now that {9J1, (/)..) is a solution of the system of differential equations on an open subinterval 8ubinterval S' of S such 8uch that G bi , i-I i - 1,• ...• .•. , n. {For (}i~or example, exanlple, one po88ible possible such luch S' is o E 8' S' and f'.(a) (/),{o) - b,. (a A).) For any a E E S' such luch that [a - h. h, a + h) C S 8 we can h. a + h}.) (0 - h, find a solution (';1, 1/1,,) systeln of differential equations on (,/I&, ... ••• , !/I ..) of the system «I 1/1,«1)" ~,(a), i ... == 1, ... , n. By uniqueness (a - h, a + h) such that !/I,{a) - (/)i(a), (Corollary 1), '/I,(x) r\ (a - h. h, a + Il). h), SO 80 that we !/Ii(:r;) -== cp,(x) (/)i(:r;) for all xES' xes' r'\ (/);'S and !/I;'s toðer to get a solution on can put the CPits "'"s to(ether 011 the open interval S' U (a - h, extrelnitietJ of 8', S'V h. a + h). Choosing a close to the extremities S'. we see that we can extend (cpl, SystCUl of differential equa«(/)1, •.. ••••, ((JII) (/),,) to a solution of the system 8' by a distance h at either tions 011 on the open interval got by lengthening S' S; otherwi1f8 otherwise we extrenlity, renlain in the given interval Sj extremity. provided we still remain can lengthen 8' S' up to an all extrelnity extremity of S. Ilepeating Repeating this procedure will give us a unique solution on all of S.

vn

f .. Corollary 3. Let S S C R be an open interval contai1ling containing a and let ft, fl, ...• • • · ,f,. be continuou8 t (x, Yl, ... Yra) E E"+l E,,+l : xES 1that continuow real-valued functions functionB on (x. •.••, Y .. ) E BatiBJy Lipschitz Lipechitz conditiona. conditions. Then fOT for any blt •••• , ••• ,J b aaliajy b.... E R there exist unique /unctions CPI: 9Jl:S-+R, tn«:h that fOT for eod& lunctiona 8 -. H, ... ,(/).:S-+R J cP,,: S -+ R auch ead& ii = I1,..... ... , n UHl we IlCllle (/)/(:r;) f,(x, 9Jl(x), have ~/(x) .. a: I,(x, ~l(X) • ... ••• , .,..(x» ~,,(x» on S and 'Pi(a) ,.o,(a) -== boo bit First suppose that this has been proved in the special case that

Is. 11,

•.. ,I.. , /. are bounded 011 the subset of E ..+l given by I (x, hi , ••• , btl) : xES I. ... E,,+l bl •...• J• Then for any ai, lit E S such that al < a < ai, at, the functions /1, ... ,/. , Ira are

bounded on the compact subset of E"+I Era+l given by I( (x. (x, bll , •.• , b.) btl) : [al. a,] a.] I, I. 80 so that there is a solution zx E (41, aolution of the given system of differential equations with initial conditions (ai, lit) equatiOIlS conditiolls on the subinterval (aI, at) of s. S. By CorolI, if we choose different al. lary 1, ai, a, the solutions we get will be the same on the intersection of the two intervals (ai, lit). a2)' Since any point of S is cona2) of S, we thus get a unique solution on tained in some subinterval (ai, at) all of S. Hence we may suppose to begin with that It, f1, ... •.• ,, ffft.. are bounded (x, btl,, ••• ••• ,, bra) btl) : x E 81. .. , on {(x, st. Let MER AI E R be such that if xES, x E 8, 1/1, Yl, ••• ..• , yy", 81, .... '1, · .,,.2. z,. E Rand i = 1, ... , n we have If.(x. YI, ••• , Yh) - fi(x,.21, •••• .2..) 1~ M «YI - .21)2

+ ... + (Y.. -

.2.. )1)112.

1.1.

l'ID. SVCCII881VJI APPROXIMATIONS VIII. Bt7CCIJ88IVJI

Let hER, h > 0, be such that 8 contains the open interval (a (0 -- h, A, 0/I + h) A) and hM Vn < 1. We shall show that given any open lubinterval subinterval of 8 of 2h there exiate exists a unique solution of the 8Yltem system of dift'erential differential equalength 21& I,(z, 1/1, n, on the subinterval having arbitions 'l/l 11/ - /,(z, 7/1, ••• , '1/,,), JI,,), i ... 1, ... , ft, trarily prescribed values of 7/1, JI.... at the center of the subinterval. '1/1, ••• , 1/ Granting this and reasoning 88 aa at the end of the proof of the preceding corollary, given any solution of the system of differential equations with initial conditions on an open subinterval 8' of 8S Buch such that II (I E 8' S' and given a E 8' we can get a solution on the interval 8' V (a - h, a + h), any Ot U (0 A), provided this latter interval is contained in 8. We can repeat this procedure 8, Thus Thua we are reduced to provinl proving the to get a unique solution on all of 8. C888 where 8 ... (a - h, is 88 aa above, all (0 A, a G + h), M ia corollary in the special case AM Vii < 1, and there is. number A E R luch that 1/,(z, bt, "" .•. , b.) I
vn

1:1I

U - (s, 1/1, ••• ,1/.) ((Z,1/I, .. ,,11 ..) E E"+l: E-+I: Is - al al < 1&, h, 1114 111' - bil

< Nh, NA, i-I, .. •••", "I. nt,

Then if (S,1/I, (2:, 1/1, ",,11 ••• , ".) == 1, ... ..) E U and i = ' '" , 11 n we have

I"(z, 1/1, 111, •' ,• •, ,,1/,,) I,(z, bl,t , "" b,.) I • • • , b.) I/.(z, II,,) - /,(z,

~ M( t

(11' (1/4

01-1 4-1

M·)tlt < M(nNW)lIt 114')'" M(nNW)I/l -

hMNVn, AMN'\I'"n,

implying

If.{z, 111, I/,{~, 1/1,

vn + A.

" 1/.) 11.) 1< I < kMN hMN Vii • •, .,

fl •..., ..,1., ,f., a, /I, bt, We can now try to apply Corollary 2 to the present 8, S,II, b.. and U, taking N NIl == = ... N .. = ... , b. ••. = == N" == N. All that is wanting for Corollary 2 to go through, thereby completing the proof of Corollary 3, is that the inequalities

lUx, 1/1, ••• , '1/.)1 h~ld hold

~

N

for all (z, 1/1, 11,,) E U and all ii-I, 11II 1, ... , ,ft. '1/1, ••• , '1/.) n. But theae these are valid if

AMNVii+A SN, hMNv"i+A ~N.

vn

Since hM laat inequality will be guaranteed by taking AM Vii < 1I the last AI(I - AM hM v"i), N ~ AI(1 Vii), 80 we are done. laat result is the following An almost immediate consequence of the last main theOrem on systems of first order ordinary linear differential equations

2:'

... (X)'l/1 + ... + 144.(X)1/. U(.(z)y.. + "4(X), Vi(Z) , ii-I" .. , n. ~ == 14U.l(X)Yl - I , ...

13.

DlrnUN'nAL _VAftONI BQUA"ONa DlrnJIIIN'I'I4L

lit 119

Corollary 4. 1.161 interval conkli,u.,., containing lite 1M point ,. G aM lor IAt S C R be em open ifltmlal _lor each aM II, v. be c:cmti"uou continuoua reoktalued ,.I-tJGlU«l/Uftdiom on 8. e4Ch i, j == .... 1, I, ... ••• ,ft ," let Uii Uq and lunt:tiou em Then aUt Jundiom fII: 9'l: 8 --. R R.•••• ••·, TAm lor (lny ony b b"t , ••• ... ,, b. E R there . . , tmique unique Ju'ffd.il1M cp,,: , " 11M have IP.: 8 --+ - R aUch aUc1a tAat t1&ot for etJC1& e4Ch i-I, ... ," _ 1aaH

",'(s) f/J/(z)

1: L• Uq(S)IP/(S) "'1(2:)fI'l(~) + 11,(21) ,,(s) 1-1

on S and ",(0) ==. b,. ~I are bounded on 8, .y 1~(2:)1 S JI for all aU If all of the function. functions Uij say I~(s>l ~ II zS E S i, j -=- 1, ft, then if S z E 8 and lilt Jlt, ... ••••, !h, -., ...... •••• fa E R R 8 and all i,i I, ... , ", Ifa, -It ·wehave 'we have

I(I(t

1-1 1001

Uq(s)1I1 Uu(Z)YI

,,(s) ) + II.(Z))

- (

UiJ(s)~ + 1I4(s) t Uu(z)~ ~(z)) I )

i-I Iool

S S M. hav~ a unique 8Olution lIOlution on the union 01 IUOh open intenala We therefore h&v~ of all alIlUGh interra1l (til, (GI, GI), CIa), which is 8.

""I

Higher order differential equatioDl equations are equivalent to ayatemI IYItemI of &rat. ftrIt, equatiODl. For example, letting order differential equations.

cP1J d'1/

dy. dr.

~. ~

-..=r'

111 .' tlz1' ••• , II. ,. --~, 1/1 == - 1/, II, 1/t 1/1 =- . ' 1/1 --= d.r:"

,," order differential equation the ft"

,z, t, ....

~ PI 11, .5.. tJ-:"') ~ Zl) tb- -1(21. - •. J1, . ' • • •, ~ Byltem of first order differential is equivalent to the syatem diJlerential equatica equatiODI

~-,. !n-w.

• -t-..

dJltt-a _ 1/• dtI-a -II.

•a1; :

1ft, ...,,.). ••• , ,.). - '(s, F(z, Ifl, Yt,...

Thua the next two corollaries are immediate oooaequencee Thus coDllequencee of the theorem last corollary respectively. l'eIIpectively. and the lut

1,.

190

't'DI. YIII. 8VOCII8IIIQ SVClCIIIIIIR APPIIOXllU.TlON8 APnoXDIAftONa

Corollary ,eal-ttaluetl futtdilm function on em an opeR open lUb.et tub,et CoroIIGry s. S. LtJ Lee I/ be CIG contiftuoua toMrIVOUI reaHrcaluetI oj the poW (fI, e.-t>. Suppoae 8uppoBe that If lIOI.iajia aati,ji,a G (I of E-+I ",,1 IAol IAtJl COJdaiftl COIIIcIiu 1M (G. Ce, Co, ••• , e...s). lApdtt. CO'Adititm, condWm, that tAot u thera there aiatI II E R aucI& aucA that tIuJt UpdiIII aiIU MER lJ(s, 'I, I/(s, '1,

... (z, al. ai, ... ...)) 1 ISM «111 - ,il ••••, 1/.) II.) - J I(s. • •• , .. S M (WI as}1t

'1. .... ,.)

·.. + (II(U.... + ...

",)1) 1'1 .. )1)111

(z, the giveR ,iven opeR open Id. TAm (z. II, ..• , J/.) and cuad (z, (z. '1, al. ... . .••,z.) a.) are in 1M Nt. TIaMa aucA tAer, mila and onlll only OM one /uRClioR Junction aucI& that there aialI one cmd ,,: A, (IG + 1&) aucA tkal .,: (a (G -- II. II) -+ - R aucI& that

~ UIIaeneNr

u..r. Giaea 0, en. 1&II E R,/& R. II > O.

+

d".,(z)

(

lor all call z E

dfp(s) ~(~)

d--1tp(Z»)' "'-1 I···. d"-I,,(z»), ..-1

tp(~), ----.-' -,;sa- - / s:I,• .,(z). tis ••• , (0 h, G + + II) Il) cuad tiM ,,(a) ~(a) - Co, ,,'(a) .,'(a) (G - II, ....... - /

CI, •••• ••• , ,,( .,(..-1)(0) Ca-l. Ca, ..-1)(0) - Ca-l·

Corollary 6. LtJ (1ft open inler&ral inle,1JGl contcaining containing 1M the point a and ,. let Lee 8 C R be em Ul, Ut, ••• ,u., /UnctioM on Oft 8. S. TIaMa TAm lor for anll any .1. tit•••• ,u.. .,, be contiftuOUI toMrIVOUI real-valued reol-vdlued /t&f&t:lioM Ct, aiau function ,,: fJ: 8 ---+ R aucI& aucA that IJKJt Co, ••• , e.-. Ca-l e E R Utere lluJre . D a unique junclioR

u.,. ..

Ul.,(a-I) U.-I.p' + U.f,O -= " ",., + .1,,' ·-1) + ... + "-I'" 11

.,(a)

ad .,(a) .,(0) - eo, Ce, .,'(0) .,'(a) - CI, Ca••••• .. and ••• , ,,(_1'(0) ,,<11-1)(0) - e... C.-I-

..wt

..t ..wt is ie of COU1'II8 Tbia lJut course the main theorem on ordinary linear differential equationa. colllidered two notable apecial special equUiona. We had previously pNrioualy considered _ , namely DUnely the the differential equations CMeI, differential equations

II' " -'/(z) -./(z)

(in Chapter VI. VI, 14)

and

11"+11 tI' +" -0 -0

(in Chapter VII. VII, 14).

nOBLDII PROBLEMS (a) Draw diapama to verify that Newton's 80IviDI an equation 1. Newton'. method 01 of 101viDI 1(.) wwb if 1 / (s) -- 0 worb / is a twice dilereatiable dilerentiable real-valued function on an opeD interval U in a cbangee lip, whoee open R that changes whOle derivative is nowhere 181'0. whose 8eCOnd II8COIld derivative does not ehalJl8 "baDge sign, provided that sero, and whOle 3rt E E U is 10 80 choeen that aIao the point point, %t also Sa E U U;i indeed UDder under theae these oilciraulll8tanCl8 the eequence sequence Sit oumetaDcee Za, %I, ZI, ZI. Z" •• •• is monotonic. 12. Chapter V to prove theae facta. (b) U. Problem 12, facts. LIt. E R, a, •• > O. Show that applYiDI applyinc Newton's I. 1M. Newton'. method to the function zI - •

e

Pwa the formula %00+, %a+aPwe - ~(z. + i:). Prove that Newton'a Newton's method works worb arrr .. 3rt > 0 by Ihowine for aD'T abowiDl that then Sa %, ~ v'Ci v'G and the map sending Sz iato into' +~) is a contraction map or oils ~(. +;) (% E R : S z ~ v'Ci1. v'G1. (This method 01 or findinc equare roote roots occura 0CleUI'II in ancient Babyionian iDa Babylonian manuscripts.)

1(.

no..... no......

191

t-

3. Prove tbat that the equation COl % - • - t - 0 baa •a unique IOlution. Show uDique nat reallOlution. ShoW' fixed point theorem is applieable applicable to the function f\lllCtlon '(s) '(%) - COI:c - t that the heel COl s -I and the interval (0, /41 and thereby find this 101ution solution to three decimal places. [0, .. .,/4) placea. II' 01 of the implicit function theorem if l(x, I(x, V) 4. Find the "maximal" U, fJ y) zI 1 and (a (G,J 6) b) - (0, 1).

+" -

S. 5. Generalise the proof of the implicit function theorem to get the fonowing following resuIt: suit: Let I/ be a continuous real-valued function on an open subset of EA+. Ea.1 that contains the point (aa, (alt ... , a.., a., b), with /(01, f(aa, ...• ... , a.. a., b) - O. Suppose that 8//8V exilts exists and is continuous continuou8 on the given open subset and that 81181/ ~ ~

a., b) " O. 8y (ai, (Oa, •.. ••• , Ga,

.

Then there exist exi8t positive real numbers hand h and I:k such that there emta exists a unique function 11': I(Zt, (x\, ... GI)I + ··· ... + (z. (Zo - 0..)1 0,.)1 .,: •.• , :c.) z.) E E" : (::1 (%1 -- a.)1 -+

z.» -

hllJ < AI Iy E RR :: I" Iv - 61 < 1:1 I" kJ

such that I(xa, /(Z1, ... •.• , s., ,,(Zl, ... .•. , z.» - 0 for all (Z., •• , Sa) z., Ip(x\, (z\, .. Zo) in question. following argument into a proof of the implicit function theorem 6. Expand the foOowing that avoids the use of the fixed point theorem: Take r > 0 such that I/ is defined on the entire open ball in E' JlI of center (a, b) and radius, radius r and such that aI/a, ia is never sero aero on this ball. Chooee Choose k A: such that 0 < kA: <'; < rl then choose hA a//a" such that 0 < hA < VfI=li ~ and I/ is nowhere aero on the set V) E Et: E': Ix Iz - al < A, IY bI - kkl.J. I«z, (x, 1/) 111 -- hi sero at precisely one point of each vertical section of the rectangle Then I/ is sera II(z, (x, V) 1/) E Ei: Ix - ataI < h, kl. E': Iz A, IY - hi < 1:1· 1/•• solutions on R of the differential equation V' 31 Villi. 7. Find all 801utiOD8 1J' - 31111

vTif.

V(O) - O. (There is an infinity of answerll, 8. Solve the system Iystem 11' - 2 vTYT, 1/(0) answeres, essentially four different ones onel near zx - 0.) with eeeentially

.3

solve the ay.tern system 9. Apply the method of proof of the first theorem of 13 to 801ve y' - ", y, ,,(0) yeO) - 1, starting the RUcceuive IlUcceasive approximations with 1/10 - 0, obtI' taining thereby a power series expansion 01 the solution. tainiol

10. Modify the proof of the first theorem of 13 to show that we may take for A any positive real number less than 11M such that the open subset of Et E' on 11) E E' : Iz ~ 1&, 11 y E RI which I/ is defined contains It (z, 1/) 1% - al SA, RJ by showing IIlch an A the given fonnula formula for F defines a contraction map on all of that for 8Uch O(tG --1&, hI), not jU8t just on a ball B. eGa A, G a + A», 11. Suppose that the conditions of the first theorem of 13 t 3 obtain and that ~-J, is a real-valued function on some open interval of R one of whose extremities is a, G, 1/1 having the properties that I/I'(z) - I(z, 1/1 (x) ) and lim t/I(z) I/I(z) .. - b. Prove that '" ~'(z) -/(x, ~(z» "'(x) == cp(z) I/I(x) = 11'(%) whenever both expressions are defined..... defined. . 12. Show that Corollaries 3, 4 and 6 of the last theorem remain valid if, instead of being an open interval, S is R itself.

191

YlII. \0111.

8Vct3S81VJO IUCCDIII8IVJI ArrROXlMATlON8 API'llOXlMA'I'lONI

13. Prove '" tlmee times Provo that if "'I, til, tIl, UI, ••• , u., v II are real-valut'd functions on R that are til ditrerentiable, dltrerentiabJe, then any IIOlution of the differential equation 1/(") ,(s), - pes) ,(z) ~('" + Ua(.z)J/(A-I) UI(S)~(-" u.(s), .is .is (n (ft m) times timee differentiable.

+ ... .. · +

+

+

real-valued 14. If. Let [0,6) [G, 6J be a closed cloeed interval in R and let A and K be continuous rea1-valued funetionaoD z,,, 6]1 respectively. 11# C«(a, 6D, 6», functicms on I(A, .. 6) 6J and {(.z, {(s, 1/) ,) E E Jill BI : s, , E (a, 6]J U # E C({a, define F(#) E 0([0, 6]) by C({a, 6) (F(tJt»(z) (F(.,,»(s) - A(z) A(s)

+ J.' K(z, II) #M fUl) III K(s,~) IlJ

(the continuity of F(.,,) F(-/!) followilll following from Prob. Probe 15, 1&, Chap. VI). Show that if I1(6 (6 - (I) (z, '1/) %,"y E (a, 6] b) then F i. i8 a contraction map, and a) K K(z, ~>II < 1 for all s, therefore there is a unique" unique f(J E C«(a, O«(a, b» such that

b»

f/(x) = A(z) ,,(x) ... A(s)

+

f

K(r, y) "(11) f/(Y) fly dy K(:r, II)

E [a, (a, bl. b]. for aU zx E 15. Let (0, [a, II) b] be a closed interval in R and let A and K be continuous real-valued b) and I (z, (s, y) ~) E E' : a S ~ 11 II S ~ z ~ 61 •, functions on (a, b] %S b J respectively. Prove that there is a unique " E C(la, 0([0, II» unique" 6» such that ,,(x) - A(x) f/(z)

J:

IlJ + /: K(s, K(z, y) ,,(y) f/UI) III

for all zs E (a, (0, II). preceding problem if 6). (Hint: Imitate the procedure of the precedilll yS ~s _ , note I1(6 (b -- a) K(z, II) I < 1 whenever a S JI .z ~ S 6. To do the pneral poera1 cue, al E (a, b), the problem reducea provine the existence of a that for any 01 reduces to proviDe unique UDique 'Pi '" E E C(la, C([a, 01» al) such that ",(x) A(z) f/'I(s) - A(s)

f

+ J.. K(s, IlJ K(z, y) ",M f/'I(y) III

E (a, al] al) and the existence existenoe of a• unique fII for all s e t'I E C«(al, C((01, 6) b) such auch that fII(S) - A A(s) .,.(z) (z)

61.) for all s% E (a., b).)

1··

+ /:1 K(z, II) Y) ",(y) .,.(1/) fl" dll + /; J..0. K(x, K(z, II) 1/) filM t/JI(II) flll dll . K(x,

CHAPTER IX

Partial Diflerentiation

ThiI chapter ill ia concmlfJd concerned with exteDcliDIthe.... exteDdioctbe methoda of one-variable ditlerential differential ea1cu1. calcul_ to fuactloDa fwlctloDI. of od8 more than one variable. There an .... few cIUIeal.... difIiGultia, 0D08 once one bM correct de8niQon _eratiabili'J f. bu the ClOl'ftlCt de6nifiion of difleNIltiabillV for functions . functiona of I8V8ral aeveral variabl.. variabl•.

1M

IX. P.um.t.L Dll'nUIII'IUTION

,I. 11. DEFINITIONS AND BASIC PROPERTIES. Partial derivativ. derivatives are themeelv. themaelves a mat.ter matter of one-variable differential calculUi. It. .A. auch luch they have al....y already made their appearance in t.hie.t.ext thietext in calow_ our CliIeuMiODl diecuaaioDII of differentiation under t.he the intepa1lign inte&rallign and the implicit aleo alluded to in our dilcUllion diacUllion of differential ditlerential funct.ion theorem. They were alIo function equationt, in connection wit.h with Lipechita Lipechitl conditione. conditiooa. Let. Let \II us recall their defi. defiequatioDl, nition, restricting ouraelvea ounelv. for convenience to functiona functiont on open subeetl subeeta B-. of 8". For any poIitive aubeet U of g., B-, any real-valued poeitive integer ft, ", any open subeet point. a - (ai, function I/ on U, any point (GI, ... ••• , a.) E U and any i-I, ... , ft, ", the .illt. parti4l pGrtial deriNtiH dIriIatiH 011 all at til a G is defined to be the derivative at CIt t.he tJi of the I(al, ••• Z" ~I, •••,,4.), , a..), if real-valued function which sends ~ So into /(GI, ".,, ~l, "'-I, Zi, 41+1••• expl'e88ion I(al, /(a" ,. :t., 41+1, ••. ,,4.) this derivative exists. (The expreeaion •••, , "'-I, ~I, Zi, ~I, ••• a..) is of coune to be undentoocl understood 81 as l(zl, / (ZI. as, CIt, ••• ,,4.) COU1'l8 a..) if i - I and in like manner 811 II /(GI (ai,•••. ••. , a-l, Go.-l,~) z.) if i-ft. Note that the function sending 2:t ~ into II(Gl, (at, . , .,, ~l, "'-I, :t" 41+1, ••• ••• , a..) a.) is defined on an open aubeet. subset of R that. that con••• %1, ~I, taiDlIJ(, taint CIt, 80 10 that it makelll8D88 makes aenae to apeak speak of the derivative, if it exiata.) exists.) The

.... partial derivative of I/ at G /I(a) or .CI is often denoted I/(a)

:! (a).

(G). Thus ThUi we

can write

.

If() _ ~a) ~/) _ l' 1(Gt, ... -/(Ga, ... a.) • "(0) lim 1(0., ••. , "'-I, 01-1, So, ~, 41+1, ~l, ... ••• , a.) a..) -/(al, ••• , CIa)

" a

iZt·....

-~G -

1m

~......

•

~%I-a;

If Il(a) /I(a) exiatt exiata for each G /t on II l! (allo (aleo 0 E U we set a real-valued function It a//azi) whose value at any a E U is JS(G); thie is the ,i'" ... pGf',itJl partial denoted '1/8%" I:(a); this dIriIatiH oj J, deriNtiH 011. notationt for partial derivativ., derivatives, We remark that there are many other notationa text. Alternate notationa notationt for a/lin. al/~. none of which ,hall be used in this text.. include /.., I.., I.., and DJ, ft., DiI,

r.n -

!, -It-

analoaoua notationa notationt for" (a) the aoaloaoua I;'(G), 1~(4),

(a) being

I .. (a), (G), and I..

(DJ)(a). (DiI)(a).

11leee are often expanded to /[(Gl, .• ", CIa), a.), ~(al' ~(alr ... , a..), a.), etc., and one These Il(al, .•• iJZt ~

evtD finde finda even

I/(2:t, .. • ,z.) (ClI, ~

)

••• ,a..,

a/(Ul, ... ,v.) , u.) (CI ( tc a/(ul'au. ••• "'.. ) etc GI, .. "a., 1, ••• , - , e .•

11.

,

DBnNITIONS AND BASIC PBOPSRT1B8 PROPERTIES

195

There are clearly many possibilities for confusion and more will appear later. No systematic notation is perfect, although some sorne are better than others. The only essential is that we know exactly what is meant Illeant in any given instance. How should the notion of ditJerentiability be defined for functions of several variables? The original definition by means nleans of difference quotients (J(x) /(xo) )/(z --:1:0) %0) does not generalize immediately. iOlnlediately. A de6nition definition of (f(z) --/(zo) differentiability must be given ditJerentiability for functions of more than one variable Ibust which does more than refer to partial derivatives, for all the partial derivatives of a function may exist at a point without the function being beiDI wellbehaved there. For example, the function on ,.. EJ which has value zero at the origin (0,0) and the value zy/(zi %y/(zi + 11' 7/)> at any other point (z,1/) (z, tI) is

+

such that ~(O, 0) and ~(O, 0) exist (and equal zero), but If is not even

ax

ay By

Bz

continuous at (0, 0). It tUfns turns out that the property of being closely approximable by linear functions can be generalized and this will give us the desired definition, as follows.

Definition. Let I/ be a real-valued function on an open subset U of Ea. E". Il" (ai, •.• ..• , a,,) tla) E U. Then IJ is differentiable at Il Let (J == (aI, a if there exist CI, ... ••• ,, Ca c,. E R such that c., lim I/(z) - (J(a) (f(a) • 1/(%) 11m .

+ Cl(XI CI(ZI -

~. --

Ill) al)

+ ... · .. + c.(z. c"(x,, - tla» a,.» 11 _ o.0. ~

d(x, Il) a) d(z,

Zl, ••• , z. The Xl, x" in this definition are the coordinates of z, x, so 80 that z == = (Xl, (Zl, •.• , z.). x,,). The d denotes the metric in E·, E", that is d(z,ll) d(x, a) = ==

%

+ ... +

«ZI - al)1 1l.)1 + .. · (x" (z. - a..)I)l/t. tla)I)III. «Xl . The limit condition in the above definition is ia sometimes more conveniently stated as follows: given any e > 0, there exists a 3 > 0 Buch that z et U and d(x, d(z, tI) Il) < 3 then if ~

I/(z) 1/(%)

a (f(a) + Cl(Xl Cl(ZI (J(a)

a

Il.) as)

tla» 1S ed(z, Il). + ... ··· + c,.(z. c.(x.. - 0..»' a).

iB used here rather than < in order to include the case The symbol S is z -- o. Il. % If for any i ... I, ... ,,n n we set Zx -=- (ai, == 1, (at, ... , 1l'_I, ai-I, Z" Xi, 1l1+1, ai+l, .•• ... , tla) a.) in the above definition, we get

or

. 1/(IlI, tli-., Xi, Z" Gi+I, tli+I, ... tla) -/(0.) 0 I/(al,.... · · ,J Oi-I, · • .,, a..) - lea) - c,(z, c, (x, - 0.;) Gi) 1I = 1·~ , ==, IXi Gil IZi - 0.;1

I s,..a, -,-,

·.

1m 111m ai".i ~i".'

I

1/(1l tli-l, Zi, 1 0, 1o ' ·••.,, lJi-l, I/(at,. Xi, tli+I, Gi+l, .•• • • .,, tla) a..) -/(0.) - /(0) -Co - Ci ... == , Zi -- 0.; Xi tIi

so that I: 1[(1l) 11(0.) Ci. Thus if If is differentiable at a 80 (a) exists, and indeed It (a) = c,. .•• ,, c,. then the coefficients CI, CI, ••• c. are unique and equal to f{ Jt (a), ... ,,/~(a) J~(a) respectively.

1M 196

IX. P.urnAL •NTlATION p.um.u. DlrnJ• DlrnBBNTlATlON

The following technical lemma will prove useful on I8Vel'al several 0ClClIIIi0ne. OO~ODl.

""*'

L,mma. Let real-valued function on Oft em aft open lUNd U 01 0/ B" B- ... _ ,., ,. "."...... L« I/ be a reakalued if _ oral" only if "..., tAer. .., aiat ~ ,eaktalued u ~ AI, .•• Oft U, continuoua contiftuoua at a, aucI& IUCA tAae tAtJt ..• , A. on a E U.TIam U.T_ I/ it diifermtioble dijferentiGbk at atJ

A 1(s)(zl - at> A,(s)(s. - aa) Ga) + + A.(z)(z. A.(s)(s. - .. a.)) I(z) -/(0) - 1(0) .. - AI(z)(zl a.) + + A.(z)(z, + ·.. ..• +

lor for each i ... =- 1, ... , nft _ tD6 have 11(0) A,(tI). lur all2i all z E U. 1",. In tAw tAu CtJ88, Ct.I86, lur /I(a) --= A,(a). If /I il is ditlerentiable differentiable at (I a we have

+

a.»

Um I/(z) - U(a) + + ,t:.(a)(zi a.) + + ... _ .o. - I/(s) /[(a)(zl - all · ·· + f,.(a)(z.. f',.(a) (x,. - ..» I . 0. I1m ... d(s, a) G) d(z, Since 8inoe

«St -

+ ··· ... ++ (:r:. (z. -

des, a) .. - «Zl - al)' a.)1 + d(z,o)

.. )1)1/1 S IZI 0.)')1/ 12:1 - ",I all + + ..• ·.. ++ Iz. -- .. 0..1,I,

if we define deftne the function e: a: U -..... R by

a.»

I(z) + /:(a)(~. I~(a)(zi - a.) + + ·.. •.. + + f,.(a)(z. /(~) - U(a) + f,.(a)(~ - ..»

t{) ()

1:l:I-all+ ... +lz.-o.l ISI-",I+ ... +ls.- .. 1

s --

fS

-....

for ~ :t .. G a aDd and e(a) I(a) - 0 we have lim e(z) a(z) - 0 and

+

!(s) + ... I:. (a)(%. - .. 0..)) f(:t) -/(G) - 1(0) + I;(O)(ZI fs(O)(ZI - at> a.) + ••• + + f.(o)(z.

u.

s l -- ",I + .(:c)(l e(s)(/zl .•• + + Is. Iz. - .. + GI' + + ... a.. I)

for all sz E U. Setting A.(z) .. Il(a) ::i: ± a(z) == !l(o) f(Z)

n, with the plUl plus sign being chosen if Zi for i-I, i == 1, ... , ft, x, -the mioU8 minus sign, we get

04 tli ~

0, otherwise

= AI(z)(zl + ... AI(z)(Zl - al) + ·· · ++ AII(z)(z. A,,(z)(z. - 0.) = /lea) and lim A,(z) = /l(a) = Ai(a), which proves half of the lemma.

-

I(z) -/(a) !(x) - /(a)

For the

~.

converse, convel'88, if A I,

••• ,

A.: U --+ R are functions continuous at a luch such that

I(z) -/(a) AI(z)(zl - al) J(~) - /(a) - A1(z){ZI (II)

+ ... + A.(z)(z. ··· + A,,(~)(z. -

.. a,.))

then for ~Z E U, z .. ,. a, we have

.. (a)(z. - 0.» + ... + AA"(a)(x,, all» I d(z, a) d(x, 0) (AI(z) - AI(a»(zl I (A1(:r:) A 1(0)) (%1 - al) OJ) + ... -·· + (A.(z) - A.(a)}(z. A.(a) )(z" - .. a.))!I -~~~--~~--~~~~~~--~~--~~ d(z, d(s, a) 0) z l-oll AI(:r:) -- Al (G)1' A .\4 I )1'z.-a.l S Al(a) II~(;,+ ... + '~;:sl ~ IIAI(z) d(s, :;' 0) + + IIA.(z) .A.(z) - A.(a)! "(Sf tI)

I/(s) - (/(0) (1(0) + + Al(a)(x~ AI(a)(z~ '- Gl) al) I/(z)

II::

Al(z) ... + IA.(z) S IA A1(a)" + ··· A ,,(z) - A.(a) I. I· 1(s) - AI(a)'!

11.

DJlnNlftOU DJlftNmOQ AND BAllO

no....... no........

197

AI, ... .•• ,t A. at a, the lut laat exp. exprellion. approachea the By the continuity of At, . . . approachel sero 88 I is differentiable at tIa and ...., limit zero 88 % approaches a, OJ proving that lis abIo that flea) fl(G) -=- A.(a) for i-I, i - I , ... ,J ft. ft > 1, I, the functions AI, ••• ••••t A. appeuiDa appeariDa in the lemma Note that if n are certainly not unique.

Proposition.. oj E", Era, /: Proposition. Let U be an Gft open aubad 01 I: U.... U- R. 1//U 111 Sa diD"""" diI"""""

at aG E U, then f ia contifttWU3 continuoua at a.

For if AI, •.. , A. are 88 88 in the lemma, then limf(%) .. lim (/(0) (f(G) + AI(:':)(Zt A I(z)(%1 - as) at> + ... Iimf(%) ·..

....

80 I/

mil

..... ....

+ A.(s)(s. -

crJ) eJ)

-/(a) + AI(G) ... + A.(a) •·00 -/(a), -lea) A 1(0) • 0 + ·..

a. is continuous at G.

lOme praotical erJ.terioD criterion It would obviously be of great value to have baYelOme for the differentiability of •a function at a point. Such. Such a eli. criterioD II alonied alonled . . it by the following result.

/Ita", ...,... /tilt",....... ,.... e u.

TlNer.m. tJft . . ...",., ,.."", 0/ ... Ba. I: U -. TMorem. 1M Let U be (1ft opM - •• tial derivatilJa~, thrivGtillu I., ... ,/~ aUt esial on ~ ., -' . ., ,..., tiol ••. ,I: Oft U ........ - ON conUatIouI ... ., • TMA I iI G. Thm ia diDer"",itJbN di8t/f'efttiable at a. .

10IIII of lenen1ity genen1ity we may IllUme II... . , . bill blUlll .. Without 1018 _me that U . ia .of center a. G. Then for any z % -- (St, (SI, ••• ••••, -.) s.) E u, Ill., all of the paID. pain.

e u.

z.), (til, (ai, %I, ZI, •.•. (ai, Ot, lit, Stt s.), (%1, ••.. • .,, %a), • .,, s.), z.), (01, Sa, •••• • • ., Sa), .... (al, ... , ..... s.), (CIt, t e.) ••• , (ai, e ' -t ...." t . .) . ( - . ••• -. -,a.)

an,

are in U and 80 are all pointe of all line Hne aepnentl between .Y CODIeCUtive _tift aepnenta betwe8ll two of these pointe. Writing .

St.»

fez) -/(0,) -/(G) == (/(%1, (f(ZI, ... f(x) .•. , S.) s.) -/(Ot, - !(Gt, ZI, .... • • ., s.» (f(al,ZI, -/(a•• ..,-, + (f(01, St, ... • • .,,s.) Sa) -/(Oa,.., Sa, ... • •• , e. (J
+

+ ... .. +

,e.» e.»

,a.»

and applying the mean value theorem to write

,.,.>

-.>

/(SI, ..• ••• ,Sa) l(xI, , z.) -/(Ga,Zt•... -/(al, ZI, ••• , s.) -!t
I(al, ... , a.-I, a,,-I, s.) a,,) -I;(a., Ga), /(GI, %.) -/(al, ... , a.) "/~«(Jlt ·... ·.,, .... -"-a,1. Ea)(s. E.)(~ ~ -..), where each ~i Ei is between a, fl4 and z. fi... .. OJ •e obtain Xi (or E /J4 .. - s, if G4 -~ - s.),•w. IHOt, ft, ZI, •••• .... z.)(St z.)(Zt - CIt) a,) +};(Gt, Et,~, ... + /~(4t, 1~(4t, •..• a-a,~.)( .. + ·.. • .,, a.-a, ·f.)(-.

I(s) -/(0) -/(a) ... fa(EI, %I, ZI, ... 41) a: n(EI, • •• , %.)(ZI %.)(%1 - Gl)

a.).

1.

1.

11. II'IU.L DlrnUN'ftA."ON DlrnBIIN114110N IX. •• PAIlTlAL

Suppoae that for each % ZE U & a specific choice of EI, ~1, ... ••• ,, E" ~.. is made. Since Suppoee

ft, ··., .. " I~ f~ are continuous at a we have n,

....

lim 1~(~1, ••• ,,z x.) "(~1, XI, Zt, ... ..) == I~(a) f~(a)

....-.

lim f;(al, EI' ~t, %" ZI, ••• ... ,,x.) - f;(a) lim/~(Gl, z.) == f~(a)

...

lim/:(al, ... , CIa-I, E.)

-/~(G) .

Hence the ditlerentiability differentiability of f at a follows from the lemma to the preceding propoeitiOD, propoeition, taking cedinl AI(z) Al(z) - .n(fl, n(E.. ZI, x" ... ••. , z.), A,(x) ... Et, ZI, Z.), A.(2:) == neal, j;(01, EI, X" ... ••• ,, :rll)'

A real-valued function fJ on an open subset U of E" E- is called diJferendifferentiabll ~ Oft U (or juat just differentiable) if it is differentiable at each point of U. neceIII&I'Y condition for this is that ft, U. f is called conA DeceI8&I)' It, .....• f~ exist on U.I tinuoualy differentiable on Oft U (or just continuou.lll continUO'lUly differentiable) if Jr, ft, ... , I~ f~ tinuoulll exist and are continuous on U; thia thill terminology is reasonable re&l!Onable becaUle because the exilt lIuch a function is nec. necellll&rily ill "Y easy theorem implies that luch .ril)' differentiable. It il to live give many examples of continuously ditlerentiable differentiable functiot18. functions. For instance, any polynomial in the coordinate functions 2:1, •••. ,,:t,. z,. on E" E- is Zl, .. continuously Era. continuoualy differentiable on E". Recall Reca1I that a map fJ of any set Bet U into E- is detennined determined by ita its m com/t, ...• ... ,f.: (fl• •· ::.,f.), ponent functions II, f.: U-R U -+ R and we often write ff= == (fl. · , f.), this meaning thatf(P) == = (JI(P), (/t(P), ..•• .. ,f.(P» ,J.(p» foranypE for any p E U. If U is an open tbiameaningthatf(p) Uisanopen IUbeet of EE" and CJ a E U we know that J f is continuous at a if and only if 11, fl' .• .. •",f. f .. are continuous at a. It is therefore reasonable to define f/ to be tliffermtitJbk tJ if fl, differentiable at a /t, ... , I. f. are differentiable at a. Similarly, Similarly. fJ i8 is said to be tli6fJm&liabll II, •.• differentiaIJU if fl' ... , I. f. are differentiable. fJ is called conlinuoU8ly continuoU8ly differen~ if II•... II• ... ,,f. continuoualy differentiable. 'iablI J. are continuously &8 in one-variable calculus, a differentiable function of a Exactly as diflerentiable differentiable function is differentiable. More precisely, we have the following inl "chain Nle".

-.f:

""'HI.

Let U. U, V be open eubNl4 018", of E", E· reqectively reapectively and let J: - V, TIaeorem. lAt I: U -+ ,: V -..... R be /uftdioM. aucA that tJUJt I is /unction.. Let Ld G a E U be 8UCA i3 differentiable at a and g9 ill fIi,f~ diI~ at lea). I(G). TAm TAen , •0 I ia diDerenliable differentiable cd at GG ond and lor Jor j - 1, ...• .•.• n N ft

..

E• ,:(f(o»(f.)j(a). .n;(a) - 1: ,:(f(G»(f~)i(a) .

(g
4-1

11. ,1.

DltftlGTlONI ....D hIIO no........ noPRTI. NftIG'ftONa AND IWIIO

199

Bince , ill (J1(8), ••• , /.(0», /.(a», the lemma proved Since, II dilfereat.iab1e dUf. .tiabIe at a~ /(4) /(0) - (fl(a), earlier impliel functione AI, ... , A.: V .....R. impli. the exiatence uiItenoe of funct.ioDe Aa, •.. , each continuous tinuoue at /(0), lOob auoh that

+ ... + A.(,)(,. -/.(0»

,(J) - f(J(0» - A I(,)(,. -/1(0»

for all .•• , m the function Ii is ditrerentiable differentiable aU 'IIfI E V. Similarly SimiI&rIy for each i-I, ... /, II at ••••, B B..: -+.B, 4, a~ 0, a, 10 there exiat exiA functioDl functioDa B A , ••• .. : U , each continuous at II, luch aueb that

Ices) -/J.a) -

a.> + ... + B ..(s)(:r. -

BA(S)(Sa -

for all aU ss E U. Therefore

,(J(z» ,(J(G» f(J(s» - ,(J(o»

a.)

..

E• A,{J(s» A,(J(z»U.(z) -/.(0,» 1: (J,(s) -/,(0» w

..- 1: E• AcC/(s» A,(J(z» 1: E• Bu(s)(SJ B'I(z)(~ -

OJ)

1-1 J-l

4001 i-l

- t (t 1-1

AcC/(s»B4j(s) )(SJ - OJ)

4001

.

for aU zs E U. Since each Bu Bil II is continuous at a, CI, since /J is continuoue continuous at a, tI, and and. ainee since each A, is Ie continuoUi continuoua at /(0), we deduce that each function

t

4001 lei

(A.o/). Bu: U-R (A,on· BIJ: U....R •.

'0

II coatinUOUI ooatinUOUI at CI. By the lemma, '0/ ia M G. I Ie ill dUferenti&ble dilferentiable at o. G. Moreover, &pin ,~CJ(G» - AcC/(a» A,(J(a» and. and each (fo)~(o) (f.)~(o,) -Bu(o), -B4J(G), apin by the lemma, each "'{J(o» 10 tbat for eachj - 1, ... •.• , R" thatfor each i-I, (goniCo) (goni(G)

1: E• AcC/(o»Bu(a) A.{J(a»B'I(a) - 1: E• f'.{J(G»{Jo)i(o). ,~(f(o»(fil~«(I). 4001 i-l

4001 i-1

U the I/ and and. , of the theorem are differentiable functions the conohllion can be written in the slightly simpler form clusion

oocuion for much imprecilion The chain rule is ia the occuiOD imprecision in notation. For example, the equation 01 of the theorem ia often written in the form

...!L _ t 11.. .ll1 ..lL_tlL~ ~ a~1

w w

8/. '/4

~.

_ JOG

IX. PARTIAL PABTIAL DIPnUN'nA.TlON DIPPIlRIlN11A110N

sinne, among other faults, f/g is not a function of This is not literally correct since, Z - (XI, (ZI, ••• , :t,,). z.. ). A correct version of this formula UBing using the a notation is x ..

the cumbersome a,(fI(XI, ag(/l(ZI, ... ••• , x..), x,,), ... ••• , / ..... (XI, (Xlp ••• , z.» x..» aXI aZI

.;.. 8g(YI, ••• ,1/.) , Y..) (J »a/;(xl,a x.) ~ a,(YI, /I (( ••• , z.) == "-' L.I .:10.. XI, ••• , z. x..)) , ... ,'''' , f,~..((:tl, Xl, ••• , z. x..»a/,(:tl, ...... . .... .110. VII ZI, ;"1 ~

v.'

..

"1/'

fiX; ~

fi1'8t formula often results The kind of oversimplification appearing in the fil'8t in confusion in practice. Here is a typical example: Suppoee that that,z .. ... 2x 2% + u == :tX + 1/, Y, so 80 that. == 3x 3z + 1/. y. Regarding Regarding.Z 88 as a function of zX and that u ... that z ... = 2, but regarding regarding.z 88 as a function of :tXand Y and u we have a./ax az/a:t ... 1/ we have a./ax = 3. The explanation of this anomaly is that the sinlle single symbol • is az/ a:t ... used here to represent two distinct functions. In fact let /: gt E' - gt E' be given by /(:t,1/) lex, II) - (x, (:t,:tx + 1/) II) and let f/: g: gt E' - R be given by ,(z, g(x, If) II) 2%'+ If. II. Then the symbol. symbol, etand. etande for both f/(z, g(x, tI) (i..., (i.e., the function funetion ,) g) 2%'+ g(f(x, 1/» II» {i.e., (i.e., the function (Jg 0/), so 80 that the two UBeI U8eII of Bz/8:t a,/ax above and ,(f(:t, functione "g' and (g o/){. (, o/l'. really represent the distinct functiol18 The following paragraph, which will not be required in what f0110wa, followa, ie intended for students adept in linear algebra. We consider E" a vector i. apace formulae 8paCe over R, defining addition and scalar multiplication by the formul88

on,

(XI, ••• ,, :t..) x.) (ZI,

1/..) ... = (XI + 1/1, Ill, ••• , z. x.. + 1/..) II.) + (1/1, ••• , 1/..) C(XI, ••• , :t..) x.) .. (CZI, (CXI, ••• , cz.). cx..). C(:tl'

If U is an open subset of E" and /: U - EM is differentiable at the point aE U, define a linear transformation (a): means of the 'IE f(a): E--E'E"-E'" by meane m Xn matrix

aI, (a») (11',)'(a» (<JM(a» ;i-I,..... .. (( :~ v' i i -i-t I ... .....;i-I aXJ (a») II

(called the jacobian matru matrix of f). J). That is, if (:tl, (Zl, •.• , z.) x..) E EE" we set let /'(a)(xI, x.. ) = == (Yt, ... .•. ,1/",) , Y...) E E"', Em, with lit, 1/1, ... ••• ,,1/... /,(4)(:t1, ... ••. , x,,) 1/. defined by the matrix product

(U~;(.» (1) -C) 80

that for each i = = 1, ... , m we have 1/; 1/i ..

..

E (/;)J(a):I:;. (fi); (a)%I. Then the differI_I 1-1

entiability of I/ at a4 implies that

d(j(x),/(a) + /,(a)(x - a» ... O. . _d.:..U..:.-(z.:..;.) '.:...;/(:....;a):.....+.;....;/:....' (.:...;a);..,.;.(%_--:4)..:...) Ilim 1m -0. ....

d(z, a) d(X,4)

12. t 2.

R10HU RlOHU DSlUVATIna DJ:IlIVAnVD

101

(Note that in the top line of the formula d refers to distance in B-, while hlle it refers to distance in P.) Collversely if U is an open E".) Conversely in the bottom hllo subset of E", E". I: f: U - E'" a map, and if we have a point aca E U and a linear E- such transfonnation T: E" - E'" auch that , I (a) d(j(z),/(ca) · d(j(z) II1m

-.

+ T(z T(x -

ca) d(z, a)

a» 0 ca» .. - ,

then we verify immediately by looking at the coordinates coordinatell of I(s) I(x) that I is f'(ca). (Thus the differentiability of I at aca differentiable at a and that T ..; /,(a). and the linear transformation /,(a) f'(a) could have been defined without felOn reeort the to component functions.) Our statement of the chain rule tranalatea translata to t.he statement that the 1 X n matrix for (g 0 f)'(ca) f)'(a) is the product of the 1 X m matrix for g'(f(a» by the m X ft n matrix for /'(a); f'(ca); in other words we have transformationa the equality of linear transformations (g 0/)'(a) o/)'(ca) - ((f(a» g'(J(ca» • !,(ca). /'(0).

Thia last la.et statement atatement 11M hIuI an immediate generalization to t.he This the following neat venDon of the chain rule, rule. which is exactly analogous to the correeponding correaponding version la.et proposition of 12, Chapter V): If U, V are one-variable result (cf. last aubeets of E", Ea, E'" respectively and I: V-~ open subeets f: UU - V, g: V -E~ are funcI(ca), tions, with fI differentiable at the point a E U and g differentiable at I(a). then ggo0 If is differentiable at a and (g o/)'(a) 0 f)'(a) .. - g'(J(a» g'(f(a» • !,(ca). I'(a).

12. t 2. HIGHER DERIVATIVES. Higher order partial derivatives are defined similarly to higher order ordinary derivatives. I..et subeet of E", E" ,I: all open aubeet I: U - R a function, Let U be an I, 2, ... ,n. ,ft. If It exists exiata on U we may be able to and i, ij integers among 1, define the ii'" ij'" second order partial derivative 011 some point 01 I tJl aI a (ft)J(ca) (Jf)J(a) for lOme (fl)j(a) exists for all a E U we get a new function (I:>i a E U. If (ff)/(a) (I:>; on U. If (fa: (fi): exists and k ... 1, ... , n, ijJc'A IIftrd third n. we may be able to define the ijlr!" order partial derivative 01 fI at aI a «fl)J>'(a), «(fm)'(a), and if this exiate exists for all aU aG E E U we have a function «(faD «fD;)•.•. And 80 on for still higher order derivatives.

u.

(fa: al80 be written ~ 81 al ), which i. (fD: can also is uaually usually abbreviated

ai/\ 8Z4 aiJ\ a~

ahall refrain from u.nng, to ~. Other notationa notations for this, which we shall Ueing, aZpZi 8zpx. include DjD,f. DjDif, Di.I.I;:wl' Di.l, f;~r/' and frirl' still higher order derivatives are /"1"/' When atill in question certain obvious abbreviations are used. For example

atl iJ41

meana

iz{iy(iy( ~))).

The 1arp larp number of poIBibie pcaible higher order partial derivatives of a ofll8VenI function of IIevenl variables is ia much reduced by the cil'CUID8tance circumstance that the order of performing the partial differentiations is ia usually irrelevant. The eimpleet cue amplest C&IIe of tbia is the equation

~-~ ~-~ azB7I a"az' aza'll "'in' repeated application of which yields

ftI/ a'I

azaya. aza.yBa

_

8'1 a.a,az azarin

aU eimilar eimil&r MUlte. COUl'8e lOme and all reeulte. Of courae BOme mild conditions must be satisfied to parantee thia irrelevance of order. The conditions in the following theorem sufficient for all practical purposes. are not the weak.t weakest known but are 8ufficient purpo8tl8. We note that slightly weaker conditions have already appeared in one of the exerciaee (Chapter VII, Problem 36). exerciIeI

Th.... m. 1At lunction Oft an em open IUbeet aubM 01 E" II&al IAol conTlaeorem. lAt 1 I be G4 real-vcalued retJl..vcalued function emaleaN 1M 4 and let i, i,jj be 4mong •.. , no ft. II 11 Cll)J (Jl)J cmd CI;): (Jm mil eziat Oft em our .... "" poW G cand ,., GmotIf 1, I, ... _ _ cmd cand en open IU6aet are contin,"*, c:cmtinumu cat G4 then (JD}(G) (fm(4) - Clm(G). (JJ)S(a). suppose i "j. There is nothing to prove if i ... -= j, 80 BO we may 8Uppose pi j. Also AltJo all variables PJ"QCtl8IIe8 by which vanabl. but Zi and Zj Zi are held fixed in the various limit PIVCUDl(4) and (f;);(4), we arrive at UDl(G) (JJ):(G), 80 we are reduced to the ~ n -... 2. suppose that 1 I is Therefore we may 8Uppose i8 defined on a certain open ball in HI E' of center G 4 - (Gl, (Ga, CIt) and that (JD; exi8t on this thia ball and are conCI~); and (f;); (J;>~ exist tinuous at G 4 and we must prove they are equal at G. a. We introduce the function 4, given by

z.> 4(z) _ I{za, I{zl, za) zi) -/(za, - I(zl, CIt) --/(Ga, I(Gl,:e.) (Zl (St - CIt) (Za - Ga) Ga)(~

+1(4a, I(G" CIt) ,

defined at all pointe zZ - (ZI,:e.) (Za, z.> of our ball of center G a for which Sa Zl "pi aa, Ga, ~ ,. CJs, In the rest of the proof we restrict oUl'IIelves ounelves to such 8uch pointe z. If St

"IIIwe_ we set

we have 4(s) ZI) - .,(GI, .p(41, :e.) SI) 4(z) _ "(Sa, ,,(ZI, St) (Zl (Sa - Ga) aa) (St (~ - CIt)

HI between (Zl, zi) and (ai, (Gt, z.) XI) is Now the entire line segment in gs (Sa, z.> i8 in our open ball, 10 the mean value theorem enabl. eoables us to write ,,(ZI,:e.) - (XI f/(sa, za) - .,(GI,:e.) ,,(aa, z.> (Za - sa).,:(Et, Ga),,{(Ea, St) z.>

E. between GI for some lOme fa 4a and Zl. Za. Therefore 4(z)... ,,~(Ea, %I) 4(z) _ .,aEl, Zr) ... =

%.-a, z. - tit

Il(E.. xi) z.) -}J(El,-ai!_. -}i(EI,-~_. laE" .:I:,-Gt .1:. - tit

12.

H I " " OUlVAftft8

..

Since (Elf as) ia is in our SiDOe dle the entire line leIIIlent JeIIIleIlt in ll' JlI between (El, :1:.) z.) and (E1, open ball and since by 888umption assumption (f;>; (f~); ¢data exists in our ball, another applica.applica,tion of the mean value theorem gives l1(z) =: (f~);(fl' (J~)~(~I' EJ EI ) 11(2:) ...

a

for some XI. That is lOme tt between al and 4 6(z) EI) l1(z) -== (J{)'(El, (fm(fl, tt)

for lOme fl Ea between al /11 and ZI Zl and some lOme is ia &l8umed auumed continuous continuoUi at a /I we deduce

att between littit and 2:1. x,_ Since (f~)~

-....

l1(x) == (Jt)~(a) . - (fn;(a). lim l1(z)

ia independent of the order of the two But by the symmetry of 11 thilllimit this limit is ditterentiations. going through the diiterentiatioDl. (Thie (This can also be proved explicitly by lOin, l&II1e "'(ZI, 2:.) x,) .. /(Zl, 2:.) Zt) IllUDe argument 88 88 above with ,,(Zll ,,(Z1, z.) 2:1) replaced by ~(2:1, - /(z" /(01, /(G" Zt).) 2:.}.) That is, ie, we allG aIao have

...-

4(X) lim 11(2:)

(f;>~(G). (f~'(o) •

In the following theorem, which ia is a version of Taylor's theorem for functions of several variables, we ahall shall find it convenient to use a uditter"differential operator" operator" of the type 8 Cl~+~+ .. , +c..._ +c..~, Cl--L+~+ ... _.

8Z1

8z. ax.

82:1 8z1

Ci, ..., ' , " c. c.. E R and for any Here Cl, any real-valued function / on an open subset of BE- on which all the first fint partial derivatives ft, . , " I:' of-/ 01' f exist, we Bet let

I:, ...,/:.

I

8

CI-'(,CI82:1

8Z1

8 \, 8/ 8/ + \4' -= CI Cl 8Z1 a/ + ... + c.. ~ 8J + ... ' .. + + c.--L c.. 82:.JI8~. )I a~l + .. , + c", 8z.

-==

c~ c~

+ ..·..' + c../.., c./:",

functiOD on the same Bame open subset subeet of E-. E·, Similarly another function Sitnilarly if all the 8eCond second partial· derivative8 derivatives U~)j ~)i exist cxist on this open Bet partialset we can (~an define

)1

8 CI + ... +c.. aza8• J/ (( CI-+"·+C.8z1

'*,

82:.

(Ci..!.. ' .. + C...!..X(Cl..!.. + '"... + c...!.. \~), - (ea--L c....LX(C1--L +".--L \~). h, + ··· b. 8Z1 8z.JI, Bzl az. 8Z1 ax.',

Similarly for hilher iterates of

..., + c. -'-, _8_. Cl-'- + .. 82:. az.

CI-'hi 8Z1

One verifies

immediately the explicit relation

+ ... c.--~-)·/ =. ==. ---_....~'l ~pl_· __••.. ." + c",--~)'/ 1:-1 •...• _ ~.c.•. . 'c..c.. --__ax---. (Cl~ hi 8:1:.. .. ....... -1....... a.l: •• i)X,a· • • aor iJza iJx.. (JX•• 0

-o

o

(I.I:•• •• •

IN . .

IlL PAIft'IAL PAB'I'IAL DI""'UlI'I'I4ftON DlrJ1DUMTlAftOH IX.

TI&eerem. Let U be an open subset 01 of Ef: U - R be a Jwu:tion TIaeorem. E'" and let I: junctilm tJlZ o f " partial derivGtirJel of order n + 1 triBt aUt and are continuoua all 01 tMote derivatwes olorder continuous on Oft U. T1aen il if a .. b.), and 1M TAm - (a., •.. , a..), a...), b == (b., (bl , .•• ••• ,, b...), the entire line aegment segment beCweM 0a and aM b are in U there eziBtB u.:iIU a point e on tAat between Oft tAw tAil line aegment segment auc1a aucA that I{b) I(b) -/{a) - f(a)

-M«b. - a.) ,:. 8:1 + ... + (b. - a...) a!. ~ )r )1)(0) -M. + ;1 ;1 «(b.':')'f)(a) «(h. - 8:1 + ... + ~ )'/){O) + ... + ~I«(bl ~I«(b. 8:.8:1 + ... + a:.t!. Yf)(a) a )"+1) fI .+ (n + « a.)h"7 + ... + +

({bl - al)

(h. - a..)

a.) al) ,:.

)(a)

(b., (b". - a..) a...)

- a.) al)

(b. (b... - a..) a...)

' 1 1)1 8 (bs h. 1) 1( (bl -- a.)

Ji)
8 (b., (b. - a..) a...) h. fb:.

(e).

Define a map h: R-ER - E- by h(e) - (a. h(I) (al

+ (bll --

a.)t, ... , a. a".

+ (b. -

a..)C), a...)t),

dift'erentiable, h(O) - a, h(l) - h, b, and h rnape 10 that h is differentiable, mape [0,1] [0, 1] onto the thie line 1eIsegline eegrnent aegment between a and h. b. Since U is open and contains contaiDl this f 0 h is defined on lOme open interval in R ment, the composite function 10 [0, IJ. The function fI is differentiable dift'erentiable since it has continuous containing [O,IJ. fint partial derivatives. By the chain rule, rule,! dift'erentiable and we have 6ftt I 0 h" ii is differentiable

•

(Jo h)'(I) h)'(t) - L/Hh(I»(h, I:Jf(h(t})(b, - Gi) fJI)

'_I '.1

- «(b. - al)

8:1 + ... + (b. -

a...)

~ )I) (h(t) )

or

+ ... + (b. (b.. -

«{bl - Gi)-' al)_8_ + (J 0 h)' A)' - «(b. Repeating this, for

II p

8~1 h. ... I, ... , n + 1 we get = 1,

«(b..

a".)~\~) \;) 0 1&. h. a..)~ 8~. J'. az.~.

.. , + (b. - a..)~)'f) a...)~)·/)oh. + ... 0 Ia. az. az. 8~1 h..

(Joh)C.) a.)_8_ (J 0 h) (0) - «(b.l - a.)-'-

By Taylor's theorem

(Joh)(l) _ (Joh)(O) (Joh)(I)

for lOme IOnte

+ (jola)'(O) (Joh)'(O) + ([oh)"(O) (Joh)"(O) + ... 11 21 (j (II' (0) (j (11+1)(1') (J 0 h) h)(·l{O) (Jo0 A) h)C·+i){r) + nl + + "I + (n+l)1

r between 0 and 1. Hence the present theorem, withe -

h(f). A(E).

The special case n = = 0 is of particUlar importance and is often called the "mean value theorem for functioDl functions of several variables". It states that there exists a point c on the line segment le(lMent between a and b such that

I a. 13. feb) --/(a) lea) I(b)

= (b l

-

at> al 8/ (c) a.) a%1 aZI

IMPLICIT rvNC'I'IOIi rVNOTION .....,.... TIdOU..

_ . .

·.. + (b. - ~(c). + ... az. az..

case the proof is eepecia1ly especially auy: euy: we If we wish to prove only this special cue just 1(1&('». nul 'Ibis jUllt apply the ordinary mean value theorem to the function 1(1a(t)). proof shows that in the present special case the bypottu.ea hypotheses may be weaklpecial cue continuously differentiable difterentiable on ened somewhat, IOmewhat, for all we need is that 1 I be aontinuously some contains all pointe of the line 1IfJIIMDt. aepnenf, between lOme open subset of E'" E" that containl G continuOUl on •a lupr 1...... CI and b, except po88ibly poesibly 0a and b themeelveI, themselveI, and aontinUOUI let containing a and b. RBIIABIt. In applications of Taylor's theorem the _ U .. REMARK. t,he let is often •a 110 that it if' ill lJR(·flll hall, 80 UAf'(ul to know that 1M entire enlirt line liM ..,.., .",."., . . . . . ,., No painta baU (open or cloI«l) cloud) if w iU ~ alrem.itiee en or, pointe oj 01 E- iR ill contained in a giv~ given ball in 1M the baa. (41, ••• ,,Ga) ball. ]t'or I~or if the points are the distinct points (/la, a.) and (b ll•, ••• renter of the ball is (Ca. (Cl, •••• c.) then for U17 any 'E R , b.) and the l'enter •••, c..) • •• ,b.)

(d«a, + (bt (bs - atl', a.)t • ·...• G,. + (b. - o.)t), a.)t). (Cl (d«OI .. , 0. (Cl,•••• •.. , c.))1 c.»)1

•

E G4' - ee)l, 0,)1, - 1: ((.j .. + ('(II," -- .." '-I w which can be written aCt - 13)2 fl, 'Y., E a, ellCI > 0, and fJ)1 + 'Y for certain a, fJ, thil clearly attains its maximum on any cloeed this cloaed interval of a R at one of the extremities.

I S. THE IMPLICIT FUNcnON 13. FUNCI10N THEOREM. simplify the following To sinlplify followinK exposition. exposition, if Z % -- (ZI, ••• , ~ s..) E Bra B- and 11 =- (Yl, (YI, •.• ••• , Y.) y) the point (ZI, Y Yrt) E E E" we denote by (z. (%, 1/) (%1, ••• • • •,, %a, z., fl, ft, ••• •• •tt ,,..)) EE·+ft. EE-+-.

LeI. m, n be poaitive Theorem. fA positive integer., integers, CI tJ E 8-. Ea, bEE-, b E Ea, and W/., 1ft II, •.. ••• ,/. , I. continuoua real-l'Olued real-l'Cllued lunctiona ... be continuous function. on Oft an open BUbM aubIet of 01 ...... IJ:-+- ".., U&aI . conIcI$tII 1M b). tDitJa tDitJ& II(a, b) "" ... -1.(0, eM point (a, b), == ..... /.(0, b) -=- O. o. s.,... 8."",.. ., ".. /fir /fir . .. . i,j-I i,; -1,•... ... ,n

rs, ··., ,.) - ClC):'" Cle):'" e~ aAd and if iB continuou. continuoua on 1M ,iven giMa open de. X n,,~ ~ ~ZiIta 0fHJ" ..... ""*" and GfttI IIaGe U&al11&e. a/,(ZI, ••.• %a. ,., ••• , ,.) _ 8/,(:,;1, • .,, :I:.,

lJfh B1h

det( :: (a, «(J,b»b») i. not zero. uro. Then there exi.t eziBt open aubMB is BUb.,. U C 8-, VeE-, V C Ba, vilA aG E U and auc1a that there exist, eziBt3 II a unique bE V, IUCh Uft~ furu:liqn ft,mdion ,,: tp: U -.... V ..... twA IMt IAtJt I,{z. .p(z» rp(z» - 0 lor eocA eacI& ii-I, n aM eacI& Z E U, and , lAiB !i(X, :: 1, ... , ft eocA % arttl auc1 IUCA . ".., lAiI function lunction VJ " i, ill conlinUOUll. continuoua. :8

_ -.6

DlI'nBIIN'I'IA'I'IOIl IX. P.urft4L PAJrI'IA.L DJITBUNTJA.TlON

The proof is a straightforward generalization of that previously given for the lpecial m ... == "ft .. :II: 1. We begin by defining real-valued functions special case '" Fit •..• E-+ra on whichla which /1,•..•• ... ,!. ••• , F. on the B&me same open subset of E-+I. are defined by F,(z, ,) - "", F,{z,

•

E C,J/(Z, I: Cel/(Z. ,), 1-1

where (CUI4.1-l•..••• sueh a& (cClI,.I-I•...•• are certain real numben numbers to be determined in such manner that Fit ••• , F. F• .. tiely the following basic buic properties: Fl•...• satisfy (1) ia continuous (I) each P, F, and each : : is

(2) for each i-I, ft, F,(4, Fi(a, b) - b, I - 1•.. ", ft. (3) , ft we·have : : (4. (CI, b) .. 0 (8) for each i,i I, i -- I1, , ... 1ft

(4) for any %, == 11,••.• ... I, n if and only z. 11JI we have !,{z, Ic(z, 1/) JI) - 0 for i .. if F~~, :II: 1. 1, ... n. Fc(z. '1/) JI) - ", JI' for i -= "'1J ft. :II:

Note that properties (1) and (2) hold for any choice of the cils. For property (3) to hold we need

III. at I: E• ~(41 ~(tJ, v,. b) -

I_I 1-1

VII'

6~, bit, I, I,"k -- 11,•.•.• ... , ft n

,~ is the Kronecker delta. where ail: delta, equal to 1 if i -:II: k, 0 if ii .. '" k. Those ThOle who see· in the last equations the statement that the know linear algebra will Bee' ft

(CII) times the X ft matrix (C'l)

ft

Z;

X ft n matrix ( : ; (4. (CI,

b») isia the n X n ft

ft

matrix. 10 80 that (CIi) (CCI) is to be taken to be the inverae identity matrix, invene of the second aince its matrix. which indeed baa an invene sinee matrix, ita determinant is not MI'O. sero. But CCI'S may a1IO aIao be found in a more "elementary'" uelementary" way by noting that for the c'J" get "ft linear equatioDl equations in the unknowns Cit. •••, Cia and any fixed Ii we set Cil, CII. ell, •••• solve these equationa equations for Ca, Cit, CII, we can Dve Ca, ••• , Cia c.. provided the determinant is not of the coefficients i. not. MI'O. zero. But this determinant is that obtained from . the square array (

Z:

(4, (CI,

b») by first interchanging rows and colwnns, colwnll8,

and we know that this new determinant baa has the same saIne value as &8 the original one, which wu given to be nonsero. CiI'S may nonzero. Thus e,ls 01&Y be found such 8uch that (3) holds. As Aa for (4), it is clear that JI) "" hoIde. that. if IC(z, I~~, 1/) == 0 for ii "" = 1, ... , ft then F. (s, w) -- " IJ~ for i , ... ,,ft. ft. To prove the converse we must Dlust show that F.(z,JI) I -- I1,

•

CulAZ, 7/) JI) - 0 for 11, ... JI) ....... .. (z, 1/) JI) -.. O. if I: e'I/J<.z, i-I, · .. , ft, then II(Z, /1(%, 1/) == ••• = == ,1,,(%,

E 1-1

For those familiar with linear algebra, this is an immediate consequence of nonaingu1arit.y of the matrix (eli). (cu). Those who prefer to reason otherwise the IlOIUJinKUlarity Ua, ••• , Uta u.. E R 8uch such that may note that we can find Ul,

13. ,3.

t !!:~!J

....1

(lJI'

rUNCTlON TRIIO.... TRIDOUII IMPLICIT PUIIC'l'ION

101 lin

(0, /;(Z, 1/), j ... = 1, .•• , ft, fl, (a, b)u. b)u" -= !J(z,

t-.l """

IOlving a ayatem for this involves solving system of ft linear equatiODB equations in ft " unknowos, unknowns, which is poISible equations baa DOuero nonzero determinant detenninant poIBible since Bince the system ayatem of equatiODB

:r;. b») this enables usua to compute

det ( : : ((a, 0, b))

. j

..• Cij all (a, b)u. =- 1:•" 4wu, = Ut, • CiJIJ(z, y) .. 1: oo - 1: ~/1(z, 1/) ~ b)u.. ~u. = Uo, 1-1 i-I

1j.l:-l ••-1

By" iJy"

,,-I "-1

which in turn implies that each 'J(z, y) ... == O. This completes the argument /J(z, 1/) that eils Cf/'8 can be found 88 as desired, 80 that we may take for granted in the Fa, rest of the proof that F •• 0', F. have properties (1)-(4). I , ..• Choose b&1l in E-+- of center (a, (0, b) Chooee some lOme r > 0 such 8uch that the open ball and radius /1, ... , I. I.. are radiua r is entirely contained in the open set on which la,

!:':

defined. Since each aFl/iI1Jl (0, b) - 0 we may aFJByI is continuous continuoua and : : (a, 888um8 80 small 8mall that for each i, j - 1, ... , ft . .ume r, taken 110 fl we have

II:: ::: I< 2~. i",1

at each point of the ball. We further 888ume 881ume r is 80 80 small amall that the continuous tinuoua function det ( :

) is nowhere zero on the ball, this being po88ible poISible

Bince this determinant is not sero zero at (a, since (0, b). Chooee Choose AI Ie 8uch such that 0 < I: k
-V ,1 -

+ <111 (1/1 - b.)1 b..)1 + ... ·.. + (1/. (Ya - b,.)1)1It b,.)I)l/l = «d(z, 0»1 + (d<1l, b»l)lIl < (hi + kl)lll == «d(x, a»1 (d(y, b»I)l/l lel)l/l < r,

b» ... a.)1 d«z, 1/), tI), (a, b» == «Zl - 41)1

+ ... +

(z.. (z. - ca.)1 0.)1

(z,1/) radiU8 r. If &lao E" and d<1l', that (z, ,,) is in our ball of radius also 1/' ti' E Ed(y', b) S k, then by the remark at the end of the last aection section the entire line segment in E" between 1/ Ey and 11' y' is in the closed ball of center b and radiua radius k. For our F, is a differentiable fixed z, each ,1, ditterentiable function of the last ft n variables OD on an 8Ubset of EE" containing the latter closed ball. Thua open subset Thus for each i-I, i .. 1, . , ., . J ft, several-variable venion veraion of the mean value theorem (which immediately the 8everal-variable preceding theorem) implies the existence of a point 11" follows the Precedinl Vi on the Buch that line segment between 1/ 11 and 1/' y' 8uch

80

1/) - F4(Z, F,(z, 11'> ... - WI <1Ia - V Va)~ :: (z, 1/'') + .. .· + (1/. - V.) :: F~z, ,) ~~: (z, V')·

_ . .

PAIl11AL DIFFERENTIATION DlrFEIlEN11A110N IX. PARnAL

Thus

I F,(~, 1/) F,(x, u') 1/') I IF,(s. fI) - Fi(z,

+ Ir. 11/. --1/' .. 1'\ ::~ 1+ ... + r'.I-\

S 11/, - r'll-I 11'11·1 ::: 1/'') 1+ ... SIrl :~: (X, r'')

I

1/'') 1 (X, r'')

11/1 - 1/'11 + .. 11/.. - 1/'" 1/'.. I) I> S _1_( ···. + 111" 2ftl 1111 - ,,', I 2ft' S ,111ft - 1/'.11 S 2~ max 111/1 (Irl - 1/',1, r'll, ... ,Iv. Il.l)

in

S

d(r, 1/'). y'). d(1/,

Therefore d«FI(x,1/), Fft(x, (FI(x, 1/'), Fft(x, d«F1(x, y), ... , F.. (z, y», ('I(X, y'), ... , F.. (x, y'») '"" «FI(x, y) .:... ...:.. F FI(x, y'»' (F.. (x, 1/) y) - F.(z, F .. (x, == «FJ(x, + (F,,(z, 1(x, 1/'»1

+ ... ··· + . (d(~.,:") }iCY, y'). S (n -( d(~;:") ),Y" sS }l(Y,

y'»I)'" 11'»1)'/

YY"

Also

y»,

d«F,(x, 1/), ", F.(x, F .. (x, 1/», b) d«F,(z, tI), .•••', S d«F fI», (Fl(x, b), "', " ' J F.(z, d«F,(x, y), ••• , F,.(x, F .. (x, 1/», F.(x, b») 1(x, 'II), F.. (x, b», b) d«F,(x, b), •.• ••• , F.(x,

s

+

1

.

Iek

k

Iek

< -j-d(Y, j"d(y, b) + "2 2' S "2 2' + 22' ... k.le. < ell

Thus the fixed point thcoren\ theorem i. i, applicable to the closed ban ball in EE" of center b and radius Iek and the map which sends any 111/ in -this 'this ball i·nto into (F,(z,1/), JI», (Recall that zx is fixed.) This (F,(x, 1/), .•. ... ,, F,,(z, F ..(x, 1/». Thi, gives giVel UI us the exiltence existence of a unique gEE" d(g, b) S k and F,(x, ,1) g) - f1, g, for i-I, i - I , ... g E Eft such that d(11, ••. , ft, - 0 for i-I, i - I , .... d(g, b) < ~ k by that is hex, /I(X, fI) tJ) .. , .,, ft. n. (Notice in fact that tI(O, i8 valid for each % xE U the last inequality. That is, 11 E V.) Since this is we can define our function f(J: == 11, and to complete the I{I: U ---+ V by ,,(x) l{I(x) =proof it remains only to prove that f{J I{I is continuous. I{I can be deduced from what has already been proved. The continuity of f(J I{I continuou8 at some a' aame To prove f' a/ E U, for any e• > 0 consider the .me this section, with (a, b) problem as in the statement of the theorem of thi8 replaced by (a', b'), b,), where b' - .,(a'), l{I(a'), and each!, each!. replaced by its ita restriction to the open 8ubset ·. subset of E"'+ft E'"+" given by

E"'+" : % dey, b') h') < .t· el. I((Z,1/) (x, 1/) E Ewa+" ~ E U, U,1/11 E V, d(1/,

:£

Note that our choice of r above guarantees that det( : : (4', (a', b'»

~ 0,

13.

IMPLICIT ,.UNCTION J'UNC'I10N THZORZM TaIlORSM

109

all the analogs analoKB of the original hypotheses hold. Analogous to U, V, II' f(J we obtain V', U', V', 11", fJ', where U' and V' are open subsets of U and V respec~' is a function 11": ~': U' --+ V' such that for each :e z E U' we have tively and 11"

80

"(x,tp'(x» f.(x, 'P'(z» tp'(x» == = 0 and d(f/J'(z), d(tp'(x), b') < t. By the unique/l(X, \O'(x» = == ... ••. = == /,,(x, ne1!8 property of 'P II' we get that cp(z) tp(x) == tp'(x) x E U'. Therefore neu f/J'(%) for all % d(tp(z),
show that if U and V are chosen suitably then II' We first sho\v f(J is differenU and V as in the conclusion of the theorem. Then the tiable. First choose V Vas continuous function on U V whose whoae value at sx is det (

z:Z:

(s. (x, tp(s») tp(x») is ie not

some open ball in E- 01 of center a. zero at a, therefore nowhere zero in 80Ille CI. It ..therefore therefore suffices to prove that tp II' is differentiable at any point zx E U at which this deterlninant determinant is not zero. Hence it sufficee to prove" differentheorem tiable at a, under no further conditions than those given in the tbeorenl E·e....... ) may be considered an 1&-tuple .tuple 01 of and corollary. Now each point of E"CM+a) E-+·, the coordinates of the first point of E-+" being the fint points of E-+", E-" beiDl (m + n) coordinates of the point of E,,(a+a), Hcond E·e ....... ), the coordinates of the .econd point of E"+· E"'+Jt being the second (m + n) coordinates 01 of the point of 8"c......., E"e-" ,.) such luch etc. Consider the 8ubset subset of E"c",+,,) E"e,.+.. ) consisting of all pointe (,1, (Zl, ••• , 1") subset of E-+" on which II, / ..... ,/.. are that each "z, is a point of the open 8ublet ... ,I. defined. We have a (tontinuous E'tC-+,d whose ('ontinuous function on thi8 this subset of E"(-"+'" (Zl, •.. ••. ,zft) , , .. ) is det ( value at (ZI,

("») :~: (z'»)

and this function is not sero aero at

Zl, ••• ," , I" are all sufficiently «a, b), (a, b), ... , (a, b», hence not zero if .', near (a, b). Thus by restricting the set on which II, ,I... are defined we /., ... ,/

may &IlIlume Zll in this set we have det ( assume that for any zt, Zl, ...• ••• ,," det(

:~ :~; (I'») (I'») ,.~ O.o.

as in the conclusion of the theorem. Without 1088 01 of &enerality Let V, V be 88 pnerality assume that V, balls in 8-, E-, E" respectively, with eenten centers we may 88Sume U, V are open ball. CI. b respectively, for otherwise V may be replaced by an open ball of center b a, of center G a that is entirely contained in V and U replaced by any open ball 01 of sufficiently IImall E-+- .: zx E U, small radius. The set of pointe I (z, (x, II) y) E E-+": 11 E VI segment between VJ has the property that it contains the entire line segJllent any of its tp .. (f/JI, ••• f/J.) where each fIJi ita pointa points and (a, b). Write II' - (tpa, •. • ,,11'.) tpi is a real-valued function on U, 80 so that for any :I:x E U we have ,,(.1:) tp(x) .. - (f'1(:I:), (tpa(x) , Uwehaveji(x,tp(X» ... ,,tp,,(x». tp,.(x». For each ii= == 1, ... ,,nandanyxE n and any %E U we have!i(%, tp(z» ... == 0, 80 by the several-variable version of the mean value theorenl theorem

III 110

U. Dt. P&JnUL PAItI'IAL DlrJ'llalltftUTlOIi DI~'l'IOIl

o --/~s, fl(s» -/,(a, I~s, .,(s» - 1,(0, ,,(a» ,,(0» - a%1 aI, (-') (SI - os) + aI, (-'> (Sa (-?(ss + .. " .. + + ~(-?(s. as as. l

a.) 0.)

+~(I')(9'l(s) + ... •.. + +~(-?(.,.(s) + ~(.,(".(s) --bs) "J.) + ~(-,>( ba) Sri Bw. ...(z) --b.) 'Ill III. for .me -ament be\ween IODle .. on the line IeIJDeIlt between (s, .,(.» .,(s» and (a, (0, b). For eaoh we choaIe cboaee apeci&o speci60 ", ••• ,". , ,.. Since det ( : (I')) (-? ) .. ,. 0, the eyatem I)'8tem s E U w.

%:

offlequatloDl offtequatioDa

Zi

+ ·.. ... + :. (I')(",,(s) (-?(.,.(s) - 6.) b.) _...!lL(,t) (z, - as) ••• - 'I, 8/, (-?(s. (a4)(. -~(I')(s, os) - ... Is, as. Is.

: . (")(fIl(S) (I')(",(s) - ha) bs) -

a.) a..)

eanbeeo1vedfor9'l(s) -bs, ... -b.. Weget,foreachi.:canbelO1vedfor",(s) -ba, ...•,,,,,(s) fIa(z) -ba. Wepttforeach'~i, ".(s) - b, - Aa(z) AQ(S)(SI, as} + + A,.(z)(St A,..(z)(Zs - .., as) + + ... •.. ",(1:) (Zl. - aa}

where eIoCh IICh Ac.<s) A~s) ill the quotient by det(

!!%i (..;)

... ,ft, + A.(.)(s. A..(s)(s. - ca.), 0.), +

(-?) of a& apeciflc apecific polypol,.-

various part.ial partial d.-ivati". derivativ. of II, ... ,I. evaluated nomial expreIIion in the variOUl .. "I. at various poinu pointe ,I, ", ... ••• ,'-, , ... Since lim .' - (a, (0, b) for i-I, ...• ••• I "ft and the

.t.... -

-

partial derivatiYel /., ... ••• ,I. were .-uned derivativ. of I., IllUmed continUOUl, continuous, the 'YanoUi varioUi Au'. are continUOUl ~, ...... All'. continuoua at G. Go The lemma of 11 ,1 impli_ implies that "., ... , "" are etUrenntiable ThUl "fJ iI differentiable M .. Go '11lUl ia clitfelWltiable differentiable at ., 0, and we have completed that" diffenmtiable for suitably auitabb' ohoIen choIen U, U,V. the proof tbM " is etUrerentiable V. HaviDI ill ..,. . how fA) Havinl proved "fJ ditrereati&ble, ditrerentiable, it is eM7 fA) . see to compute the

*.

ftIiouI W. YUioua partial derivati""~. derivativ. We apply appI,. the chain nat. rule to fA) the equatiOlll equations

to_

f,(s., ... , ... .... ",(:e), 9'l(s), ••• , .,.(.» ".(s» - 0 /-<SJ, ••.,

topt,

:£~

(s, + :: : . (s, .,(s» ::: (s. ,,(s» tp(s» +

+ ··· ... + +

z:Z:

.,(s» ~ - 0 (s, ,,(s»:;

(equivalently, lor for anyone likely to be confUled confuaed by the a notation, (f.)~s,.,(s» (f.)~.,tp(s»

+ (f~~I(Z,"(Z»(A>J(z) (f')':"'l(S,.,(S»(.,s}J(s) + ... + + (f,)4.(s,,.(s»(.,.)1(z) (fi)~(S,,,(s»(,,,,)J(s) -

0).

6xed j and varyiJlI varyinc i we set pt •a ayatem of ft equations in the For an,. AIl1 fixed ft linear equatioDi

B"",

•ft UD1mcnmI unknowns .... "",, ••• , ''''', the detenniDant determinant of this I)'Item ayatem heiDI being bs 1st 'SI

det ( %i %: (s,(s. ,,(s») .,(s») .. ,. O. Thus we candv. can solve explicitly for the varioue various ..;-. __• det( In cIoiDI dolDc tbiI thia we . _ . the d_red delired information infonnation that under the conditions conditioDi of the OOI01lary 'tpJbl is continuous on U, and this completes the proof. 00I0llarJ each IICh '"e/Isl

13. • 3.

DlPLlClT TBBOUII IMPLICIT I'UNCftON ruNC'ftON TBIIOUIl

211

, gil) be (I a conCOrollary 2 (InHraeJunetion (InveraeJunction theorem). Let g ... (gl, (Qt, ••. • -.,1111) tin'U01.dl1l function Irom an open aubaet the tin1l.OUBly differentiable ,'Unction subset 01 E" Ea which conUJins containa tM each g. being (I a real-fJalued real-val'UBd l'Unction subset, and point b into Ea, etJCA function on thia open aubad, ;III

IUppoae auppose

'i

det «g.)j(b» "0 (that ii, ia, det( det ( : ; (b») '" -,I: 0). Then there exist open mbaeta aubseU U, V of Ea, E-, 'WitA with (thot

be such that 9g is ia defined at each of V and tM restriction b E V, auch etJCA point 01 r~striction 0/ g to V i. Junctio11 g-l: ia a one-one map oj 0/ V onto U whose inverse l'Unction g-I: U --. -+ V is continucontin",oualu O1J.8l1l differentiable. diJ!erentiable. On the open subset of EtA }JIll consisting of all aU points (x, (x,1/) 'II) such that Ea and 1/ E EA y is in the open subset of E" on which 9g is defined, we define functions /1, ... ... ,III functions/I, J III by

z ~

I.(z, y)

-

z, - g.(y).

g(b). Applying Corollary 1 to 11, ••. ,, III I. and the point (a, b), we Set a G -- lI(b). /1, •.. get open IUbeeta subeeta U1t , V such that there let VI1 of Ea, with aG E U Ull and be VI, 8uch exists a unique function ,,: tp: U,-+ Ul - VI such that z - ,(\O(s» g(tp(z» for all z E Ul difterentiable. The map" map tp is one-one from Ulonto U1onto and tp is continuously differentiable. .,(Us) tp(U.) - ,-l(Us) g-I(U.) f\ fl VIVI' By the first proposition of Chapter IV, g-l(U g-I(UII )) is subset of E-. an open subset of the set on which 9g is defined, hence an open 8ubset Ea. Us) is an open subset of E". 1, V = == f,( u1), Therefore ,,( tp(U.) Ea. If we set U == = U Ul, ,p(U 1), then the restriction of 9g to V is one-one onto U; U j furthermore the inverse g-l: U -+ _ V is just fP, tp, which is continuously differentiable. function ,-1: :II

«g,):) -- det( The determinant det «gj);) det (

Z;~; )

jacobian deter) is called the jtWJbian

minant (or jGCObian) jacobian) of g. It is frequently denoted

a(gl,· ..• , gil) gA) a(gl; a(1/1, ••• , y.) a(YI, • • •, 1/.)

or J,.

The inverse function theorem implies that, if 9 is a& continuously differentiable difterentiable function from a& connected open subset W of E" Ea into EE" and g(W) is a connected open subset of the jacobian of g is nowhere zero, then ,(W) E- and ,g is one-one on some open ball centered. Ea centered at any given point of W. n -- 1, ,is g is indeed one-one However, II9 need not be one-one on all of W. If fa on all of W, for then the jacobian "g' is either positive or negative on all of W, SO 80 that ,9 is either strictly increasing or strictly decreasing. decreasins. But if n ft ... == 2 (r,8) E» the "polar coordinate map," which sends any (r, B) E EJ such that r > 0 COB (J, 8), is not one-one, although it is one-one if 9(J is into the point (r COl " ,. r ain lin '), 211". restricted to any open interval in R of length 2....

III

IX. PABTIAL PABTIA.L DIrn:RBNTlATlON DlrntlIlNTlA.TlON

PROBLEMS -+ R liven given by 1. Show that the function I: B' JlI-+

,) - {{

I(~, I(s, y) -

IZI1I,1 lsl~IYI

if (z,,) ,. (0,0) (s, y),.

o

(s, 1/) y) - (0, 0) if (%,

is continuous. Where is it dift'erentiable? dilYerentiableT it 2. For which real numbers a > 0 is the function I(s, r) - (:cI dllYerentiableT !(z,,) (zi yI)- differentiable?

+ .->-

B' .... ..... R that I: 8'

given by is liven

3. Show that if I/ is a real-valued function on a connected open subset of 8" 3.8how B- and fa -I: I is i8 constant. I: - J: - ... --/~ J~ - 0 then 1 4. Let I/ be a8 differentiable dilYerentiable real-valued function on the open ball in E" B" of center (01, •.. radiu8 , and suppose that J~ I~ - 8//8z" allaz,. - O. Prove that there (GI, ••• ,0.) , ca.) and radius, is a unique real-valued function , on the open ball in B-1 of center (01, ••• , Ga-I) G.-I) and radius' radiu8 , such that l(zl, I(zl, ... , %..) z") -- ,(:rl, ,(Zl, ••. , :r-.I), S.... I), and this tbiI (Ga, , is differeDtiable. dilYeren tiable.

I be 8a real-valued function on an open subset U of 8-. 8". Prove that lie I is 6. Let J continuously dift'erentiable dilJerentiable if and only if there exist continuou8 continuous real-valued functions AI, AI, ••• ... ,, A. A .. on the set

I(ZI, (Sl •••• Z., fI" y., ...• 11.) E E'" B'" : (:rh (Zl, ••. , :r. · · · ,II.) J

~). (wI, (rl, ••• ••••, rJ 11.) E •••• • .,, Zw),

UI UJ

such that lUoh

I(s) --1M A I (s.1IHzl I(s) l(r) -- A.(:r, 1I)(zl - 11.) If'>

A.(z, 1IHs. + ... + A.(:r, ,,)(z. -

f.) II'>

for all aU %, s. "fI E U.

8. Ut U be an open subeet 6. Let subset of R, let a, ~: 11: U U -+ R be differentiable dilYerentiable functiona, functions. let subset of B' V be an open sublJet E' that for each 11 E U contains the entire line segment y) and (11M, y), and let I: J: V ..... -+ R be a continuous between the points (a(y), (a(I/),I/) (~(y), JI)t function such that ai/By allay exiata exists and is continuous. Prove that if F: U U ..... -+ R is defined by F(y)

then F'(1I) F'(II)

il.C.'_c., a1/ay

~ J(·)~

.c.)

(x, 11) dx JI) dz

~·' I(s, y) dx, =- 1.c.l

+ fJ'(II) --/(a(y), 11) a'(y). a'M· +/1({J(1/), ~(11), 11y) fj'(y) / a(II),' ( )

(

)

normed vector spaces (Prob. 22, Chap. 111), 1I1), let U be an open 7. Let V, W be Donned I: U .... ..... W diff"",iable subset of V, and let Ga E U. Call a function /: dijJ"entiablt oh at a if there transformation (Prob. 22, Chap. IV) 7': T: V -+ exists a continuous linear transfonnation ..... W such that INch III(z) -/(a) 0)11 - /(a) - T(z - a)1I I·• IIJ(z) 1

~ ~:

liz-ali

0

- ·.

(a) Prove that if I is ditrerentiable diIJerentiable at at a, then T is unique (so (80 that we may write T generaliaing ~t wu was done in the Iaat , 1). wri~ 2' --1'(0.), /'(a) , pneralising Jut paracraph pancnph of '1).

noaL1llll no .....

III m

(b) Prove that if IJ il then I ill iI continuouut continuoulat .. i. diRerentiable differentiable at •0 thea (c) Prove that if W - B" t.hen ia differentiable at 0a if &Del oaI7 0D1J IIif the ~ com" Ulan I/ ill ponent a' o. CI. pone1lt fUDctions fUDctioDl of I are differentiable at (d) Prove the following pneralisation chain rule: U H V, W, Z are pneraliatioD of the cbaiD nonned NIpeOnormed vector spaces, apaceI, if U and U' are opeD .eubeeta .bIeta of V aDd W n.peotively, and if /: i. dilerentiable ditrerentiable at the poiDt point 0CI e U u aad aacI ,: U''''' U' .... Z I: U ..... .... U' I. is 0 I 18 differentiable dilerentiable at 0• &Del c,. CI 0 n' n'-ill dilerentiable differentiable at /(a), I(a), then, thea ,oils ,'(/(0» /'(0). ,'(1(0» 1'(0).

8. Verify Verily that if

",~: R .... ..... R are twice dilerentiabl. differentiable fUDCtiou, fullctio.... II If a• e ., a, aDd &Delli ",1/1: II

/(%, I(z, tI) JI) - ,,(x ,,(z - ay) aJl)

9.

+ '/1(% (s,,) then I/I(z + ay) aJl) for lor all (z, JI) e 1fI, /lI, theD

~ ~ ~,~ ay' 8z'· a1/' - Go'ax" Verify y) - ,-""'/ r-"'"/ v'i .tisfi. .times the differential equaVerily that the function u(z, JI) tion

,.. Ih h au aza- ",. Bzt-

a,·

Do the arne J(t)trC-'11I_,,-III.tlt, where I., 18, b) is a• cloeecl eIoIed .me for the funetion /.. /.' I(t),-c-"II-v-mtll, R and /: I: (a, b) continuoue. interval in Rand bl .... R ill iscontinuoua.

10. Show that if f/ is function on OD all an opeD il! a continuously contiDuoully differentiable. real-valued 'unction iD E' B' and Bt//8zatl iJll/aza, -- 0, 0. then there ....tiabIe interval in there. are contiDUOUIly continuoualy dil dilerentiable real-valued functioftl I. on open intervala in R B IUCh IUch that functioDl /., fa, I, intervall iD

!(z, ,) -/.(t:) ,.<71). I(z, r) - fa(z) +Ih). 11. Prove that if U U II! is an open ball in Eand II.... B" ad such that differentiable functions IUch

. __. 1.: /.: U ---t R are contiDUOUIlJ contiDuoualy ..

Ph. Ni .M aXj aXI

8%i aZ,

lor all i,j i,J - 1, I, .. ...0', ft, then tbere for there exiI!tI exists a Iunction function ,: U .... R IUoh euch that /, ft. (Iii",,: (cal, ... , a.) ca.) il ia t.he the can_ ceotAJr of the ball, fl - 8F/8z aF/az,i for lor i-I, ... , R. (iii"': If (alt /t' by try defining 1-' 0

F(z.. ·... z.) = F(:.c., · .,, z.)"

J:

f.(I, J: MI,

fit, ••.• a.) dt Ga, • •,, 0.) lit

...

I,(za, t,',81, fit, ..., ••• , oJ + J: /I(ZI, sJ ", 1M

J::..

+ /'"., !,(Xh at, ... , 0..) .. (z" ... , z-a, S.-a, "I) ",.) ell.) I.(z .. Zt, ZI, t, ',at, a.) dt + ... + f.-II !I.(z" funet.ion /: I: E' B' ..... .... R liven 12. Show that the function given by

+,.

fez, r) __ {%' za +". IIif (z, r) ,. / (z, ,) (s, r) ,. (0,0) (0, 0) ~. ~.

{

o

if (z,,) (z, r) -- (0,0) (0, 0)

ill continuously differentiable and has is baa all ita its eecoad eeeoad ordaorder partial part.iaI cla+fUivee, ct.ivUiYel, but that

~ (0,0). ~ (0,0) aza, iWrI (0, 0) ,. " ~ itIfz (0, 0).

11. lit

IX. IlL PAJrIUL P.&II'IUL DlrnUN'ftAUON DlI'nUIftt&'I'ION

13. Let on an open aubeet subaet of ]II E' containiog containing the point I& I be •a real-valued rea1-valued function fwactioD OD (CIa, its t.bird third order partial derivatives and (Ga, 01, lit, Ge). ae>. Suppoee Buppoee that I poueeees poeaeeaee all ita that thee these are continuous. Compute (XI -Ga)-I(ZI - GI)-I(%I - 1It)-I(ZI tJ2)-I(za - 1It)-I(/(zl, Ga)-l(/(Xl, ZI, Xt, Za) z.) lim (ZI eq......,)~( ..,........, <-1...... ea>

-/(0., %I, Sa) -/(ZI, lit, CIt, Za) x.) --/(ZI, Za, Ge) GI) -/(Ga, %to Zt) -/(zl, /(ZI, ZI,

+ /(Sa, /(Zl, lit, CIt, Ge) + I(AI, /(01, ZI, %1, Ge) Ga) + /(AI, /(GI, lit, 01, Za) %I)

ae».

- /(0., I(Ga, as, lit, CIt».

14. i)~.

+ :: ( V zI + "., (v JJ', tan-I tan-I 'IJ/z), y/z), and the

(b) U. UII8 part ::: part.
,,0

I, where/(~, where I(s, r) - (', I) (Here 11 v - '0/, (r, ') Laplaeiau Laplaciau is to be expressed expreeaed in terms terma of the partial derivatives of , with I.) respect to ,r and B.) 1&. aU terms of 01 the multivariable Taylor formula if m - 3 16. Write down explicitly aplicitly all and "tI -- 2, coUeoting terms together po88ible. ooUeotins t.erma topther wherever poIBible. All opeD opeo aubeet of B-, gr.,/: - t R. R a differentiable 18. I& Let U be all I: U U--. dilerentiable function. IUDetio... Prove that if IJ attains a maximum or a minimum at the point 0 E U, then/{(o) tben/{(o) - ... I~(II) bas all ita its aecooo second order partial derivativea derivatives I~(A) - O. Prove convenely conveneiy that if / has continuous I~(G) - 0, then the restriction of /, to lOme some continU0U8 at IIA and "(0) /((o) - ... ••• - /I.(A) open ball of center 0 attains a maximum at 0G if the "R X" X 1& symmetric matrix this matrix i8 is (( (J~i(a» Compute 8(z, B(z, 'I/)/8(r, r)/B(" I) if sz - , COl I, "r - ,rain sin I. B. B(:I, r, .)/B(" I, I,,,) I sin ", (b) Compute 8(s, II, ,)/8(r, fII) if z - , COB I sin ain 'P, 7JfI - ,r sin Bin 'sin 'P, and

COl'

.-,coa". , - reos".

I(z, 11,') r,') -- 0, then 19. "If 1(2:,

8a,••. !!r. _ ~ .• 8z Bz .. a, 8:r, 8z arBzB'

-1 -1 II " .

BeD8e out of this thia nODBeD8e Make eeD8e nonsense and prove. cloaed subset IUbeet of ga gr. which contains the entire line segment between 20. Let 8 be a closed pointa and let I/ be a continuoUllly any two of ita points continuously differentiable map from an open aubeet of Bgr. containing 8 into gr.. opeD E-. Suppoae Suppose that f/(8) (8) C 8 and that there A: < 1 such IUch that is •a real Dumber k

i; E («/j)j(Z»1 {/i)j(:r,»1 S I: k '.1-1 ~.i-I

Prov. that the restriction for all zZ E 8. Prove restrictiOD of / to 8 is a contraction map, 80 that flxecl point theorem is applicable. (Hi"': the fixed (Him: You may want to use Prob. Probe 20, Chap. VI.)

CHAPTER X

Multiple Integrals

In our treatment of one-variable intepation integration we were primarily continuOUl funetioaa. fuo9tiOIML.,.. ~ concerned with continUOUl

Step functionl proof., but their WIe" use '" functioaa appeared in the proofs, W88 eaai1y avoidable f.4lchnical ~hnical device. In multiW1III an euUy variable integration we are of COUI'II8 course still primarily intereated in continuous functiona, interested functions, but the neceity neceB8ity figures with curved boundaries for integrating intepatinl over fiIurea lorcee (airly pnenl pneral noncontinuous noncontinuoUi funcforeea us to conaider ccmaider fairly tions. In thiI this ohapter we .tart with a atraiptforward tiODI. atraightforward mimickinl of mimicking 01 what was wu previously previoualy done for lor one variable. At a certain point the need for generality entails lOme pnera1ity entaila apecial coDlideration consideration of leta of volume aero, but this special i8 quickly pueed. We end up with .trongel' atronger hurdle is reaulta than before for the one-variable caae, cue, in addition speci&cally multidimeuaional to all the eBI8Iltial e8IeIltial 8pecifieally multidimensional etatestatementa.

116

X. IlUIIlIPLa JNTIIORALa 1IV1/lDL1I11IftGa&LB

11. 1UEMANN RIEMANN INTEGRATION ON A CLOSED INTERVAL IN P. B". EXAMPLES AND BASIC PROPERTIES.

Recall that a closed interval in E- ia is a sublet of EE" 01 of the lorm form aublet 01

E EEft : G, tli S Z, for each i-I, i =- 1, ... , tal, ft) , 2:, S b, bcfor ••• , Ga, numbers IUch luch that a.., bt, ba, ... ••• ,,b" b. are fixed real numbera

(SI, ... {(2:I, ••• , Za) z,.)

< ba, bt, GI < interval the cloHtl cloaetl i1&IenHJl iftllmHll .,..".irUJtl tl«erminetl ", bf/ GI, (II, ••• , a.., Ga, ba, bt, ••• •.. , b,. "(II, ••• t a., a.., bba,1, ••• ••• ,, b. and we note that the numben Gl, are themselves themaelves determined by the closed interval. The notion of an opeb interval in Eft is obtained in a limilar sinlilar manner nlanner by replaciDl replacing each E- il 8ymbol Iymhol S by the symbol aymhol <. where

••• •.• ,, tit. a..

til, (II,

< < b,.; we call this thia closed

DeJin.idoR8. bt , ••• ••• ,, b. btl E R, H, with (II 01 < bll , ••• , a.. ca.. < bb•• Dfdira'doJu. Let Gl, (II, ••• , a.., ba, •• By a partition oj closed i1&lervtJl interval 1 C E" E· determined by GI, ••• , a.., Ga, 0/ the cloaed bt, intervals [Gl, (01, bal, btl, ba, ••. ••• , btl b. we mean an n-tuple of partitions partitioDl of the closed intervale ••.".,, (a-, [a.., btl] b,.) in R, that is, iI, an ordered set of n finite aequencea •• sequences of real numben N1 ), (ZIO, 1,2:1 1,, ••• 1,2:-/, N1), ••• ,t (z,.o, (ZIO, Zll, %11 %,1, ZI', ••• , 2It Z.NI), (%.0, z,.1, St.1, z,.', ~IJ ••• , z,.N.) Z.N.) (2:1',2:1 ••• ,, Zl 2:1NI), (2:1°,2:1

IUpencriptAI are indices, indicea, not exponente) (where the 8Upencripta exponents) aueh luch that for each

ii-I, .. 1, •.. ... , fa n we have (I, 04 -

2:' 2:1 < < 2:1 < ... •.• < zl < z~ ~~ < < 2:~' zl" - b,.

Th. tDidtA tDidIA of 01 this partition i. ia defined to b. The be

(2:1- 2:rl1 : i .. 1, •.• N,I. max (z/-zr --1, ... ,ta ,ft and i-I, j -1, ... ,,N,). If / is ia a real-valued function on 1, by a Riemann RiemtJtata IUm /or lor /J corrupon.di,., corrapontliftll to the riven given partitiota aum 1M partition we mean a 8um

1:

.

/(111"""·, ... • •• , '1/./.'''1.) !bJ,'····I., ,.1"·'4) (2:J!' (z~. -

I.-I •••••N,; ••• :i.-I•••.•N. ;I.-I •••••N. 1.-1•.•.•N,;•.•

2:l/rl) Zt/r-I ) ••• (s,,l· - z,./.-I), Z-/.-I), • • • (z,./.

'1//1"'1. E (zl-l, [2:101, zi'J. 2:/1). We ... y that I/ ia where each 'll1a···J. aay is Ricmman Rimumn inlegrtJble integrable Oft 1 exilts a number A E R lueh if there exist. such that, given any ft > 0, there exists exist. a lueh that 18 - A 1< a6>> 0 such 1< ft whenever 8.ia 8.is any Riemann lum 8um for /I corresponding to any partition of 1 of width Iell less than '; in this thil cue case A ia is called Rimumta integral intflgrtJl 011 all on Oft 1 (or OlIff' over 1), ia denoted /'/. the Riemann I), and i.

J,I.

il n -= - 1 we have exactly what we bad Note that if had in Chapter VI, except for a Blight change of notation. For ta n > 1 we have an immediate waS done earlier. The ahove above partition of the closed ('lOlled generalization of what was lubdivides 1 into N"N interval 1 in B" E- subdivides N tNI ... •.• N. closed aubintervala 8ubintervals no two po88ibly at extremitiea, extrenutiee, that is points pointe (2:1, (ZI, ••• ••• ,, z,.) s.) of which overlap, except poBBibly

11. II.

b,, b

. . /"'/., " "

BlallAHN . IIBIB1IANN IlITIIG&\ftON IIl'l'llaA'ftON .IN ...

D'r 117

r- r-,,/ r--r--

------------~-,--,-~

/~ 1;'1 1;'1 ~---~ _.L __ J._J ------;..,.!---,-t -"'--,;l-~ ...... ,," ",1 ,''' .-It'- - ----r'-- - - ,- r~ - - - -. I I , --"'- -...... -. ' " I ", ......r "," I , " f ' ,I ,.' r-h.....- -- -;; -;::.... --- r-~-;~... -- ---fGti3!~ I I II•, I, ,to ,. . . . . ,I '" .... ,,," I , , ,' I I I ,," f-1~~t==I~QI' :I : .-¥---------I '.-¥'---------I ,I .."',I ,," ,I ,,," I I .' •II .~---1-------------, . ---,-------------!'-. -r'---.....Ii--f I ","

"

........

*,,'

III

I..

",'

,

I I, I,

I

I

I

II, II

I" II

II

I I I I

I

I'

I I I 'I I

I , I , I I , 1 / 1 I I I

I,

ll

I

II

, I I ,

I

I I'

,I

'Il,''

II I'

--------------l--f--7-----------------~--~--~----T~-I Tl1 I ' I

I

I

: ,~--I----f-TI _____________ __ .L ..L __ -'____ J __ J II

I

,I'

FrOUD 34. A partition of cloeed interftl InterYal ill IttduoeI . . .bcItriIioa Into hdo . . . FrouD 01 •a eIeIed III .. . . iIacIuca IIUblnterval. no two of _lap, . . . poIIIbIy IlUblntervale 01 which 0YWIap. eIIIIIPt poIIibI7 M au...... extr......

eueh thatx. that:l:i ... :1:1 i,j, U &8 is illuetrated Figure M. For. such xi for some lOme i,i, illustrated in FigureM. Fora Riemann sum pointe <J~ eum corresponding to a specific partition, the varioua pointa 0 there exietI exiatI •a number Riemann integrals ., > 0 such COrreepondilll to • partition euch that if S 8 is any Riemann sum for IJ eorrespondilll Ieee than , (such partitions axial. then we have 18 - A I, of 1 of wid th le88 exiet I) tben 2e, and linee t.his is IS - A'l < e, U&8 a eoneequenee consequence of which IA - A'l < 2., since this true for each e > 0 we muet have A - A'. =0

4., ...,

/,f.

There are numerous numerooa alternate notations for f,f. In the case n == 1, 'naere I,

k..

1 by we have already denoted ~• ..I written

f(z) th. dx. For n > 1, /,f is sometimes L'J.'f(x) I, f,f

J,fa, •. /rfdx, or J,f(x)d(z), /rf(z)d(:r:), or J,fth1'" /rfdx1'" a d:r:". Sometimee " integral sip signs are used, as 88 in Sometimes"

/(x, JI) JJ,/r I(z. II) dx d:r: d"

or

dx dll JfJ,/r f(z, I(z. JI, II. -) a) d:r: d" dz.

tU.

cue n - 1, expreeeions "Riemann M in the cue" I, we shall abbreviate the expreuions intep'able" intepable" and "Riemann integral" to "integrable" and "integral" reepective1y. The comments made at the end of '1 of Chapter Cbapter VI are reapect.ively. apropos here. In particular, since there are other methods of integration than that of Riemann, must be exercised in collating Riemann. care moat coUating the results of thia chapter with results in other texts.

FlQVU II. 81. EumpIeI EIamp_ 01 RlemaDIl RiemaDa au.... auUIL If the functiOilI function I OIl on the pven liven cloeed clOled iIltervall Interval I FIcnIu In IfI III ... .... the value 1 a' at each point polnt OIl 011 or wlthiD within the indicated Indicat.ed oval and the value ..,. of I then aD)' COI'I'eIpOIldinc IeI'O 'at " all otlaer poio" pointe oil lIllY RIemann IWI1 eum for lor I COITeIpoDdinc Indicated PM1ldOll partlUon of 01 II II ill the IWI1 IUID 01 the areaa of or certain or the •w the lDdlca&ed certaiD of ............ reo...... Inw wblch IIII illaubdividecl. aubclivlded. Wblch Which rectaDp. rectanalet are to w be lncluded Ineluded la the ... . . . cIepeadI depeadI OD 011 how the poiD" pointe (,.,,/a, "",/a) are choeeD. choMA. For dlfrerllb't differat ill (,,"", W.,,) ahoioeI 01 u.u-e poio" pointe t.be the UIlIbaded un.haded reotan'" rectanalet are never iIlcluded, Included, t.be the eIaoioeI liPU7 Ihaded not be Included, lncluded, and the darkly IlIbtI7 Ihadecl reetaIlalet may or may DOt alway. iIlcluded. Included. ...... . . . rectanclrectanalee are alwa)'8

ou.

11.

BlBMANN .lITIIOII&ftON INTllCltiftON .N IN . Bra. IllDANN

119

EullPLll /(%1, ... •.• ,z.) C, a conatp.nt, coD8~nt, for EulO'LJI 1. If I is as above and /(s., , Sa) -.. e, x.) E 1, all (ZI, (s., .•. ... , Sa) I, then for any partition of I, say the partition in the above definition, we have

E I:

Ja ••• 1('II1 J., ••• , "il···I.)(stl J/.il...ia)(zlit - Sll-I) Xlia- I ) ... (x.l. - s.I.-') %.;"-1) • • • (Sal. /("./1 ..04,

ia-l 11-1 ••••• ..... N.;.•• NI;... :Ja-l••••• ;Ia-I..... N.

J1 -==i.-l,•••• Na:••.I: e(sll E:J,.-l••.•• N. C(ZI

1) ••• %./1Zli a-I) • • • (%01. (x,,'-' -

i.-l, .... Na:... ;J.-l ..... N.

-=

N. NI

,./ I: (%./1 ttl E (~IJl

N. ~

I) Sal.~i.-I)

/.- ») ») (Sal. - Sa » ... ·· ·(( LE (Z.I. x"i,,-I»)

S.il%1/1- 1I

\il-l '11-1

I

J.-l 1.-1

- e(bt e(ht - at> at} ••• (b. - a.). e(b.l -- at} Since all Riemann 8UJD1 8UIIl8 equal c(b 41) ••• • • • (h. (b" - a.), all), we have a function that is integrable on 1 and

f,e = e(b e(hll --

f,e

a.).

al) 41) ••. • • • (h. (b. - a.).

EXAMPLE I be as above, let E. El E (a., (ai, haJ, bll, and let I: J: 1 --. R be EXAMPLil 2. Let 1 a bounded function such that 1(%., ••. Suppose J(Zl, .•• ••• , :t XII) Xl pi! p6 E Ell· a) = 0 if %. I/(z., ... , z.) %..) ISM for all (s., IJ(ZI, (Zl, ... ..• , Sa) z.) E 1. Consider the partition of I1 8Upposed to be of width lellll appearing in the definition, supposed lese than a, and the i ., ••• , ",,/1"'1.) Riemann lunl !(1I1/Ja,...1., y,i a...Ja) - 0 unlfJIII unle18 lum in the definition. We have 1(111 1/,/a... 1. - EI, which can be tNe most two diltinct di8tinct NI, jl'., 10 that 1/./1'''1. true for at mOlt

I

I: E.

/ ,.. 04, ·••• 1(,11 ../1"'1.) (%1/1 /(yl/l''';', • · , "1/,/1"';') (ZIJa

II-l ..... NlI... ;Ia - ••••.• N. i. -1••.•,Nt:.•.:Ia-I•••.•

I: E

S

.

I) Sl/,-I) Zlia- 1) ••• (z.'- -- Sal..z.J.-l) • • • (z,,1.

I

I) ... 2 M a(%s1 a(zi1t - :tsI..zit-I) · · · (Sal. (z.iwa - Sal.z.i..- 1I)

h-I, .... Nli••• N.; .•. :i;;.-I ..... N. ii-I•..., - l•••.•

N.

NI ~

a( {::1 'r' (:.:sia '2 6( ~ (%s'I h-I

%..-1») ... ( .E I: (Sal. %"-1» ···( (:r;.J. -

-2 M -=

.

,.-1 1,,-1

%ala-I»

Sal.-I»)

I(bsl -- a,) •••• (h CIa). - 22M M '(b tit) .. (b.... - (1,.).

/,1f,J = O.o.

/,1'" f,J

genera1ly, More generally. = 0 for any bounded function ... 1, I, ... , Il,,, and some Ei for which there exists some i ;: fi E (ai, [ai, hi] b.l luch that /(Zl, I(sl, ... , ~.) z.) == 0 if %i f,.•• 8uch ~i pi! ~ E Clearly

1- R I: 1-+

with

EUMPLII EXAMPLB a. S ex, ai ~

3.

Let 1 be as above, and let ai, ... , a., a., fll, ... , fl. (j. E 8, ft, Define I: 8R by J: 1I

< fj, fli S bii for ii-I, n. .. 1, ... , R. I

1<-' l(zl, ...... ... , x.)) --= {:0 {

%i E (ai, fl,) if z, fl.) for each i-I, i .. 1, ... ..• , nn

E 1 and %i ~ (a., fl,) fl)i (%1, ... , z.) x..) Eland Xi e ~ (a" for Borne 80Dle i-I, i == 1, ... ,n. , ft.

'f (ZI, ••• , 1if

1, ... NI), ... Consider Z11 ••• , SI %,Nl), ••• , (z.', (z.e, z.I, ~t, ... ••• , Z.N.) Z,.N.) of I, 1, CoDllider the partition (Xl0, (SID, SI less than "a, and a corresponding Riemann sum Bum supposed suppoeed of width I... /1 •••1,., ••• 8 ~ /(1/1 , v.il···J..) (si' (~J!a - Stlrt) ZIIa-I) ••. ••• 1: I(Ytla"·I., •• ',1/.Ia"·I.) II.l••••• 11.1..... NI;••• Na;••• ;I.·l iI-~l ••..• ..... N.

(%ala - S.I.-I), Z./,.-I) , (z.1.

1,...J.) where ,1&"'1. ,1t···J. E (ZFI, (Z~l, %I') jl, ... ••• ,,j.. S., ... ••• ,, 1I. 1/,.1•••• 1,.) sic] for all i, i., i.. Since !('II.!1... I(,"·"A., /a ···I., i, ... I ., .•• ••• ,,'11. is or it i. not. not in the ,...1.) i. ie 1 or 0, according M as the point ('l (1II1a 11.11•.•. open lubinterval detennined by orl, ClI, ••• , «., a", fJI, ~t, ••• , {J., fl., we have lubinterv&l of 1I determined (s" -- S,,-I) 8 - 1:. ~. (SIJI ~1/1-') •.. • • • (z.1. (z.1.. - z.1.-') :1:..4 - 1),, II,... J11 ....,/0

.

the uterisk thOle i., jl, ... ..• ,, i. j,. for which aateriak indicating that we include only thoee ,...1. E (ai, 111/ ....(or., fJ1), fJi), ••• ,,1/,.1, ...1. E (all, (or., (J.). p., ,. 1/1 1I"J····J,. fJ.). For i-I, i - 1, ... •.. , ", ft, ohooae obOOl8 1'1,91 from amoDII, N I 80 that &rnOIlIl, 2, .•• •• ", N. si'rl ~ or. ~".' S a,

e

s~rl < {J, fJ. ~ S~'. < si", z~', S,-.l S S'-'.

Then 1/lt... 1. E (a4, p, + 1 S q, - 1 and ,i'''.J. flll...J. E (or., (a" fJ,) fj,) if ,1&"'1. (or., fj.) fJi) if p. ~ i, i. S~ ,.

i. < PI p. or i. ~ ,f'.•. Therefore i, j, :>

1: ~

(Slla - S,,-I) (s,ia ZJ!I-I) ..• · • • (z.1. (z,.l. - z.1.-1) Z..J,.-I) ~ S8

+ls11 S" -h... -11...... +1 sJsl-S .. -I .. +1 ~it $" U'ta+l Sta-l

1: ~ ..

~ S

(sal' (Z1/1 - s./r :l:IJI- 1) ••• • • • (%al. (z,.J. - %ala-I)

.. S/, SIl; I1IIoSI- S .. PI $/1 ~,t;...':"$~$'"

or

,,-I .-1 (( 1: (s.l, ~ (Z I /I -/,- ... +1 1,-..

..-1 ...-1

~8 » ·... · · ( 1: ~ (z.1. (~1. - z.1.-1» ZtaIa-1» S /.-,.+1 1.-.. +1

/1- 1l Sl/r %1

~ II

~ ( 1: S ~ (s.1a (%,/1 --

h-..

h-~

,.

,-

» ... ... (( 1: ~ (%al. (z,.1. -

I S~-I» %,h-

I ) ). %al.Z.J.-l)

h-~ 1.-..

p. and qi we have fj, fJ. -- or, ~ s~, By the choice of p, Ct, S ~i" -- sjlrl ~t3lr-l < (q, (q~ - p. p, + I)', 1)1, we must muat have qi - 1 ~ P. - II, iellese ... than each 'Pi + 1 fori for i , ... , nif n if I'is (fJ. -- Cli)/2, a.}/2. in which case the last inequalitiea ~j inequalities become

10 that we

(Z,.-1 (:1:,.,..-1 - %,,"") (ZIti - S %1",-1) ••• (%a" (:1:,.-. - S.",,-I). %......-1). (Slr l - s~1") ... ) •.• ... (z.,.-I s."") S ~ 8 S ~ (Sl" ...-I) •..

. . than I, Since our partition has bu width Ile18 a, for each i-I, ... , n" we have or, - a I < %j#.1 Si'rl a,

s~rl < sl' < si" %/" < or, a, +. + a and fJ, fl. - ,I < sl'.1 J:/' < fJ. {J. + •. I.

Therefore (fJ. - al «t - 21) •••• ({J. - aor... - 21) 2.) S ~8 S ~ (fJl (Ill · · (fJ. (/JI - or. CI,

+ 2.) 23) •.. • · · (fJ. ~-

«. + 21).

Since the real-valued function on R which Benda sends any point. point t E R into (II. a. + 2t) 2C) •••• ({J. - or. ie continuoUl continuous at 0, given Any •e > 0 (/J, - al • · ({J" a,. + 2t) is. we can find a , > 0 such that for the above Riemann sum 8 we have (fJI - all ori) ·••• 18 - (6, · · (fJ. ~. - «.) a.) I1< < •.e. Thus 1J ilis integrable on I1 and

J, J f"f' (fI, (fJ• ...: ori) ... ({J. ....: era) • • • (It.

or.). er.).

II.... JUZllANN INTIIGUftOJf . 1M .. II auNN IliftOAll'IOJf . IIEXAMPLE

4.

111

&8 above and I: 1 I --+ R is defined by If 1 is as

I( f( %1, SI,

Zit .•. ••• , s. 2:.. are rational if s., ) { 0I otherwise s.) -:II {

•••• • • , Z..

then &ny takee on the value any open subinterval of 1 I contains points at which I/ tak. 1I and al80 takM on the value 0 (thi, (tbi. ia is a • ample simple ooneeCOllIealso pointA! points at which I takes quence of the corresponding fact for ft n -- 1). Therefore both ((hi Gal ••• ••• .. - aa) (b.. correapondinl to an, tlftll pu1ition partition of 1. (II. - 0,,) a.) and zero are Riemann sume sums col'ft!lpondilll It followl followa that If is not integrable. ProJHMition. RienUlnn Iuu tM lolJtMi,., /olIotIIi,., "..,.,.u.: ~: RienllJftn inUgrotion inUgrtJtion w (1) inUgrable reokalu«l ,Ml-NltMJd ~ on Oft 1M .... ,.., ......, ....... (I) 111 and , are integrable 1 01 Ei, E" Uum O&en I/ i. integrable inlef1rtJb,. on 1I OM and

+, +,

I,

g) - 1,1+ J,g. /,1+ /".

/, U+ (J+ ,) -

(I) 111 inUgr~ real-tJtJlued reol-tJolued ~ . . .. , inIerNIl (') 11 I is ia on tJ" integrable JuftClioft on Oft 1M UN . itIIerNIl of 0/ E" tmd tJftd and c E R Uum O&en cf i, i. integrable on 1 and

/,cl- c /,1. J,cl-cJ,I. 1+,

Given any partition of 1, a Riemann IWD eum for corr-poodlDI to 00I'ftIPIJIlIII thie partition is the 8um 8UDl for I/ ~ co~ to ... t.biI sum of a Riemann IIWi1 partition plus a Riemann BUm thia puIitital, putitica, ..... _ IWD for , correepondiDi col'ft!lpondhll to thiI similarly &a Riemann sum BUm for cf cJ comepondjna colT8!pOnding to this· pu1i_ partition II ec ibneI tUn. a• I corresponding col'l'e8ponding to this partitiOIL partition. BeDel Riemann sum for J Renee the propaIitioD pIOPOIitioD is quite trivial. (Those wishing to write down a proof in all detail..., deWl . . , do. do .. by eft"ectin.luitable effectinJ ,uitable minor ebanpe chan~ in the proof 01 the 00IftIIIJ0IIdiD ~ .... ....,, of Chapter VI.) immediate consequence of the propoeitloa An inlmediate propolition ia that if 1_ 1._ integrable on 1, then

,

IN

J, (J -,) -- 1,1/,1- J". hu-,) h,· iftI.etIr"",. ,..,.......

Propoaition. 11 J u (1ft inJ,egrable ~ /tIftI:lion /w&dioA on Oft Me .... ........., ProJHMidon. 111 iI em ......, 1 01 z E I, lAM 1,1 oJ E" tiM and I(z) I(s) ~ 0 lor all aU "E /,1 ~ O. i. any Riemann IUm lor for 1 &IlJ' partition puthlon of I1 For if 8 ill I OOJllIpOndiaa oorrelpOndilll to tD .., thenB O. .. then 8 ~ o.

All in Chapter VI, there are two immediate coroUarieI. As eorollariel. Corollary 1. I. 1/1 /tmdMJAI ".. 011 '" IAe ..." eloMI III aM and , are iftUgrtJ6le integrGbk ~ /tmditIu intInIJlI and I(z) S ,(z) aU "z E 1, O&en IS ifttmHJl 1 01 E" tJf&fl g(z) lor all I, Uum I S J"o

J,

J".

_

X. IItJIJIUUI IIfftOLU..

GwollGry II I iB is Oft an iRlefrable integrable ,.I-fJalued real-flalued lunctilm on the cloBed interval Corollary J. 1// function Oft, E" IAtJt IIaot u is delmnined ddmnined by aI, CIa, bt, lis, ..• b,. and I(z) :S 1I 01 0/ EGI, ••. ••• ,, a., •.. , b. aM m :S ~ J(z) ~ M lor Jor aU zxE I,Uten all E 1, lAm

m(bs as) ... (b,. - 0.) CIa) :S tn(b. - 0.) · • · (b. S

J,I:S M(bs j,J S M(ba

as) ·... CIa). 0.) · • (b. (ba . -- 0.).

12. OF THE INTEGRAL. INTEGRATION ON II. EXISTENCE 01' ARBrrRARY SUBSETS OF 1:". Eta. VOLUME. VOL11ME.

Lemma I. A reaHHJlued ItmdiMa lORa olEreaH10ltuId lunctilm / on a cloaed if&lenHJlI inlmlal I 01 E" ia is integrable on only if, liMa gWen an" Oft1l « tMr, .uialI aucI& tAat IIaot em I if and anti Oftl, • > 0, titer, . " a "umber 1I > 0 aucA 18, mad 8S.1 ar, Riemama corrupondi", 10 181 -- 8.1 < • tDlNnever tDlacruver 881 and Riemann auma fUme lor I corrupondi,., to PfJrtitiont tha" 6. I. JIG"itimu 01 0/ 1I 0/ width tDidtA leu tJaon § 3, Chapter VI applies verbatim in the The proof of Lemma 1 of 13, present cue, case, if we change the symbol [a, la, bl, wherever it occurs, to I, and I(z)dz to J. the symbol J(z)tk

L·J..

f, /.

Deflnilion. E" is called DeJinitlon. A real-valued function IJ on a closed interval 1I of EN1 ), ••• J, (z.', a .up .,., ftm,dion (~l', ZI', ••• , Z1 /tmdilm if ~here there exilts exists a parlition partition (ZI', ZINl), (x.', Z.I, x.1, •••• N_)) of 1 IUch ~.) ••• , •~... such that for any (ZI, •.• , Za), (Ill, (J'I, ••• , JI ..) E I we have !(Zl, ... , Sa) Za) ,. '" 1(1/1, ... BOrne ii -- I1, , ... , n 8uch such that I(z,• •.. ••• , ,.) , ..) only if for BOme •.. ," St ~ .. '" ,~, 111, the cloeed interval in R whose extremities are Zt Zi and 1/4 11, contains at leut one of the points ~,Zil, ~,Zil, ••• ••••, zI" z.N'..

In oonatant on any subeet subset of 1I consisting of pointe points .In other words, I/ is CODItant Zi is restricted to one of the subsets (z' (zl,J S,l), Z,I) , in which each %4 (sl' Zit J•... J ••• , , Izl"'. tzl" ,. In particular, IJ takes (s/" J sl)••.. sl) •... ,J (sl'r(zl'r-I,I , %1"), zl"), (sIt, Iz!'. IIx,', on only .. a finite Dumber number of distinct values. The functions of Examples 1 step functions. functioDl. and 3 of 11 are Itep (Sl, ... Za) (ZI, ••• , Sa)

Le",. . J. A AIIep . , /unction functiOft on a closeJ interval 1I in A'" A.... i, Lemma i. integrable. In l , ••• , %,Nt), particular, il (~l', Za •.. , %,.N.) oj 1, if (Zl', Zl', ZINI), ••• , (Z,.·, (z.', X,.I, z.', ... Z.N_) i. a partition pArtition 01 I, if (CIa alep Junction I: f: 1-+ R ia aud& lell ••• ...lalla-I•...• h'il-I ..... Na;•••;.;.-l••••• Nl; ... ~-I ..... N. N_ C H, and if iJ the IItep H it auch """'/or anti ;, N"~ we have /(ZI, IAat lor Oft, ;1 - 1, ... , N 11;; ••• ••• ;j j. j" == = 1, I, ... , N J(Zl, ... ••• , Z,.) z ..) .. OIl ••• ...1. 1e if S sir-I Jor each BaM i - I , ... ,,ft, n, tAen then 01a r i < ~ < zi' lor

f,1 - 1.-l Cia ...Ia(Z&'1 I..(zl' f,J 1:u. ..... N.CJa••• ••••• N.;... u.-l la-l.....".;... -1..... N.

ZII&-l) ••• (z..1.. - %ala-I). Zal.-I). zlll-l) • • • (%ala

"1I .... "h.

z.

Ia(SI••.. zi' for each For if we define Ia(~'J ••• ,,~) z..) to be 1 if zi'-I zl--l < ~. < zl' , II and zero J - 1: Ci Cia•.•. ••. 1. j. is a function i-I, ... ," aero otherwise, then 1I.. f(Ji•••• f(Ji •••• i_ J••••• j. I ••... J-

that. baa the value zero at each point (ZI, ••• , Za) of I for which all the that iaequalities , ft and j == iaequalitiea s,,. Si '" ~I si hold, for i .. ... 1, I, ... ," = 0, ... , N i. By the

on I OIl

12.

.xuRI1NCI' IIlUM'IlNCIl or 0' TH. ma INTBOBAL INTIlOB.a.L

D3 123

additivity of the intesral intep'al and Example 2 of the previous section, we have /, (f (J Ci,.. os. (/'i, ...s.) = o. By Example 3 of the previous section, for Cil"'''' lI'il"'''')

I,

E L

il ...../a ,J. il•...

jl, •• .•.• ,i. ,j" we have all it,

nI,J, (/';'.. 1. = (Zt (XIII lI'ia ...I ..

.

I

1). n- 1) '••• Xlii-I) • •• (z..'" z..",-I). Zt (z"ia - z"ia-

Thus

J,I /,1

the given expression for results from the linearity of the integral, that is, the tint proposition of the last section. ie,

Propo8itJon. Junction I on the cloaed interval 0/ E- i. is Proposition. The retJl-f1alued retJl.."alued lunction interlllJl I 01 inlegrable on 1I il there exi8t .tep lunction. /I, I. integrable if and only if, lor for each e > 0, tMre eziBt atep functions fl' f. on Iauchthat and

I,

/, (J. (fa -/I) - It>

< e.Eo

The proof of this is exactly the Banle same as that of the analogous proposition in 13 § 3 of Chapter VI if we Inake make a few appropriate changes in notation. Since we shall refrain from Inaking making the precise transcription, the reader should carefully check this statement.

II the real-valued 01 E" E- is Corollary 1. 1/ real-tJalued lunction function I/ on the tAe closed interval 1I 0/ integrable on 1, tAen then it is bounded on 1. The following siolple simple result could have been proved much earlier, but it is especially easy to prove at this point.

Corollary J. 1/ clo8ed intervoJa II 1 C J are cloaed intervala in Eft E" and J: I: J --. - R ia is BUM auch that

I1(%) (x) ... eziata if il and only if il the == 0 lor all xX E J - 1, then the integral 01 0/ I on J ezutA integral uiata, in which inlegral 0/ 01 the restriction re8triction 0/ 011J to 1I on 1 exi8tB, which. CQ.3e case they are equal.

Denoting the restriction of fI to 1I by the saU1C 8&lIIe letter lett.er 1 / when 110 no confusion postJible, this corollary cOI'oUary lItatcs states Ilinlply silllply that ::> 1I and J ill is polllIible, JJ;;;= J if J :> fint note that Lelunla Lemma 2 implies zero outside 1. To prove this, first itnplies the truth of Corollary 2 if 1 that. I exists. I is a step fUllction. function. Next suppose that existllo Then for BUeh that fl(~) S f(z) I(~) S ''(x) Is(z,) fundiouslt, "1.011 on 1lauch thatft(z) any e > 0 there exist step fUIl(~tioll8/l., Eland /1) < eot. l~xtcnd ft. /. to functiol:' for each ZX E I and luch Buch that /, V. UI - It) ft, fa fUllIltiOl;' on /l(z) --I.(x) ThclI 11, I. are step fUllctiolls J by setting eettinl /l(Z) /.(%) ... == 0 if :J:x E J - I. 1"huu /1, /s on J.It(z) /(z) S Is(x) /.(z) for all zx E (J. J, l1(x) S I(x) E J, and (J. (f. - It) /l) ... (fa - It) /I) < e.

II/J Ir/r I J, Ir/

1

J,

IJ/J J, blI exists. Since J,/rf,S J,IS J,/aI. and J,J,It= bitS /JIS Thus L 11 S Ir I S J, It... II It S II IS /JI. == /,1., J,I., we have I\ 11/JI1 -- 1,/1 1,1\ S I,J, IsI. -- I,J, 11It == /,J, (f.(J. --II)/1) < e. Since IJI. E.

224

X. MULTIPLE INTEORAI,s IIULTIPLIIINTIlORAI.8

J, If = J, I.f. Finally Fi'nally > 0, we have J., suppose that J, J., fI exists. Then for any f > 0 there exist step functions Uh(II, (12 ~ lex) f(x) S ~ (l2(X) J, (gt «12 - Ul) (II) < f. g. on J such that (l1(X) Ul(X) S g,(x) for each x E J and J., the last inequality is tme for eooh ear.h

f

The restrictions of gl, gt to 1I are step functions on I. 1. By Lemma 2, (g. -- (II) g~ ~ «12 (g, -- (II), gl), 80 that (g, gl) < f. Hence If exists, and «12 «(12 -- (II) this completes the proof.

s f"I"

J,

J,

J,

It is now convenient to extend the notion of integral. First let f: Eft solne bounded subset of E". Eft. I: E" -. - R be a function which is zero outside some

We can x Eel. integral interval interval

then find a closed clO8ed int.erval 8uch that lex) 1(%) == 0 for all interval I1 of En sueh caU J integrable on E" and define if the latter We call f'" exists. Thi8 SUppoBe l' C E" ~J" is another closed This nlakes makes sense, for suppose I' C such that lex) f(x) === 0 for all % E eI'. Let I~t J be still another closed clO8ed x Eel'. of E", such that J :::> corollary, ::> 1I V 1'. I'. By the last coroUary, exists if

f.-/:= f,f f,1 f...

I

J,f f,1

and only if J,f J.,I exists, in which case these are equal, and similarly exists if and only if

f" fI

IIfJ fI exists, in which case they are equal. Therefore J, If f"

exists if and only if J" fI exists, in w~ich case they are equal. Now consider an arbitrary f'ubset C Eft and an arbitrary real-valued mbset A ACE" function/on Eft --+ R by setting function Ion sonle some subset of E" that contains A. Define]: E" l(x) == = I(x) 1(x) f(x) if x E A, J (x) =~ 0 if x eA. fl. A. We say that I/ is integrable on A 1 and define fI to be Jif the latter integral exists. This agrees with the

L

f." I."

IAfA

closed interval of E". For any A and I, J, fI previous definition if A is a clO8ed can exist only if the set of points of A at which J is not zero is bounded and I is bounded on A A;; thiR this follows from the present defiuition. if J definition. En, we say that A A htu has lIOlume, volume, and define For an arbitrary subset A C Eft, the volume 01 oj A to be

== vol (A) =

fA 1,

exists. A neceuary necessary condition that vol (A) exist if this integral exietl. exi8t is that A be bounded. If 1I is the open or closed interval in lJft Era detennined by Clio tilt ••• , tJ,., btl , ••. ••• , bfa, b.. , then vol (I) 4.) ... a.., (1) - (b. (1J 1 -- 4,) · · · (b" (b.. - a,.). ara). An oxample oX&Il'lple of a IS" having no volunle.is volume is the set of all points of a given bounded subset of Eft whose ('oordinatcs rlosed interval II of E" all of ,vhose C'oordinat.cs are rational numbers nunlbers The word volu11le, volume, as u!lt'd (cf. Example 4 of § 1). 1'he u~d here, is often replared replaced by n-dimensional volume, 1J01ume, or Jordan nle08Ure. measure. If n = 1, I, one often UBes n-dinlensional uses the InItJth instead, and if n = word length == 2 the word ClreCl. area. The two propositions of § 1 possess the following immediate ilnnlediate generaliza.. generalizationa. tions.

12.

anrraNOJI or 01' .... B1111ft'11G11AL IlXI8ftNC8 I.......-w.

ProptMltfon. PropGIJ'eJora. l~itm Integration At.u Aoa UN tAt loIlovti,.,liftearit, lollotDi1lf 1iftesril.1/ proper"-: proper": C B- and tmd I tJftfI (1) 11 II A ACEand , are i""4N6" i~ltI ,..,..,.., NGl...,.., fwdiou /ufIt:4itma 1INn/+,ia-,ableonA tAen I ia tfttegrabltl Oft A _ anti

+,

_ . .

'" A,

Oft

+ ,) -- il+ fA I + i,· fA ,. iL(fC/+,) (') 1/ C 8", I ia Oft ~ recakaltMd _ ce E R, II A ACE-, em i integrabltl ~~ /tmt:liIIrt Oft '" A ,., B, tAen cf cJ it on A A GftIl _ """ it w."abliI mt.grahz. em

Jua usual, it follows that if I/ and. and, are integrable on A then As UlUal, (f - ,) - LIL1- L,· L,. fA C/-,)

PrOfHM'eJora. 111 11 I it ia em inlefrabltl ~ /trN:ttin. /tmt:lu. .. 1M ...". __ A ProJHM'tu,n. CIR if&l4rGb" on UN E- tmd/(z) tAen fA/~ E" mul/(z) ~ 0 lor all z E A, """ I ~ o.

fA

II f, I, , Gr, are in"""'" integrabls ~ /trN:tiou /tmt:liou Corollary 1. 11 E- cmtl/(z) _ I(z) ~ ,(:e) ,(z) lor aU all z E A, """ tAen LIS E"

fA ,. LIS L,.

of 0/

'" __ A 0/ oJ Oft 1M ,..".

.

Corollary J. Let ~ IN uftIMM& '" _ 1M ......, A Aol I be Oft em mt.gr.le ~~ ~ /'/tmeliM& oJ E-. SUfIPOI' 8uppoH U&Gl I1aGt 1ft mS ~ /(s) I(z) S ~ M lor all 2: z E A_ A .,." . , A Acu ......... 7'_ TAM Ea. ".. mvol(A)S m vol (A) S

i/~JlwA(A). L/~ Mvol (A).

8ubeetl of Blero are eipeCiaD,important. IS . .iIopther ..., Subeets E" of volume 181'0 eapeciaIly important. W. W." lOme of their properties.

Prop08.don. AoW: ProfHM'tJon. Tile Tlul /ollatIIiftg lolltNirag .tJlemenU ~ Itoltl: (1) A A...". Aoa 1IOlu ... IeI"O if ,., only f, ."... , . . ..,. .., • > 0, (I) 8UIMet A 01 0/ E- Ita ttOlu. . ItIO - onlJI 1Aer. am. GII fi,ftUe finite ftUMber number 01 eloMl (or 01*') If" . ..,.. lAere aiItI oj cloutl opM) iftImJalI w.r-lt .. ita II" . . __ contcriu _ 1M tAt .... am 01 oltllhoaellOluma em&tcIiftI A fJftd tIIAoN ",."." ia it ,., leu .... lAM .. (I) Anti An1/ aubM eubM 0/ 01 GtJ aubM 0/ of BE- oJ 0/ ........ ia 0/ oJ ....... (') Tlul tlfttoR _ion 014 [mite ftu""" number 0/ of . . . of 1J't If" 11/ ia of (,) TIN 0/ (I pile "' .... ... -0/ Hlu. . . .o. 1IOlu...... o. 11 A C BB" Ita Aoa IIolWM Nf'O _ _ eM ,., BellBe .. ........ Acu ....... lAM (4) 1/ HI""" aro ... vol (BVA) (SUA) - vol (B - A) ~ vol (B). (6) B- 1aaa Au ........ _ ... Nf'O __I: I: AA ... (6) 1/ 11 A A C 1l" - R B .......... " ........... /tIrtIMIWt. ~, lINn tAen O. (8) 8 C E-l 8--1 ia compad _I: a it" conImuou., lAM SaM eM ~".,. (6) If 118 c:ompad and I: 8 .... - B ".,. 01 8-, ii.e., .•. , tile tAt Nt ,., of I/ ita E-,

fAI-O. fAI-

.

coni""",

-s.t,

«ZI, ... EE-:: (ZI, ,Za-.) E8,/(s" E8'/(ZI, ... ,~ - ..:1. (2:" ••. ,z.) ,~) EB(SI, ... ••• ,z.-.) ..•,....., Ita-.....-o. Aoa~.ro.

To prove (1) we may IUppoBe A C I, for lOme closed interval 1 of 8., B., mayeupp088 since oDly eetB have volume. Let I: 1 .... only bounded eeta - R be defined by aettilll aetting

1

I(~) - 1 if z~ E A,/(~) A,so fez) A, fez) - 0 if z~ E 1 - A, 80 that vol (A) - f,f. 1. If vol (A) - 0 then for any te > 0 there iI is &a partition of 1 luch 8uch that any Riemann for I corresponding leIS than .. comaponding to thia this partition hal has abeolute value 1_ euch Riemann IUIn sum for I correepondinl correspondiq to this But one such thia partition is ia the eum·of volumes 01 of tboae those closed aubinte"aII eubintervals of I (for the 8ubdivision IWD' of the volum. lubdivision of 1I com.pondinl correaponding to the liven partition) which contain pointa points 01 of A. Hence ill contained in the union of •a finite number of closed intervale interval8 the sum eum A iI whose volum. is 1_ Convene1y, IUppoie euppoie that for each • > 0 of wbOle I. . than .. Convenely, C II V··· V •.. V IN, where each 1, is •a closed cloeed interval in ... we can write A ell U 1., 11 iI BIWD BUm

N It

(I,) and E vol (II)

E 1-& 1-1

~ 2:

< ..

•.. , N we define II: I-R For ij -- II, , ... /1: 1-+ R by I'(~) Is
1 nIl II(~) -- 0 if S~ E 1 h1.10 so that Ell is & a step function on E 1" 11 and /I(S) 1 -- 1 I-l 1-1

If

I, 0 Sf(z) S/(~) S EfJ
sS E•" vol (11) <<.. Ie

("Eli - 0) ... t"E vol (1(I ('\ nI,) J,f, (Eli 11) N

N

Ira

1-1

/-1 J-l

Since for each te > 0 I/ is 8&Ddwiched sandwiched between the

1-1 I-l

E Ell, N

two atep step functions functiODB 0 and

fj, we know that

I-l

f, em. f, S 11 exists. Since 10 f

11 fll < • for each e > 0, we have 11"" 0, that hfSS 1 f, I:,/I for each •

0, we have /,/- 0,

1-1 I-l

Since

0

o. is vol (A) - O.

This provee dealing with closed intervals.. intervals. AJJ A. for open interThia proVf8 the part of (1) dealinl reeu1t for cloeed closed intervals rerharb that any OpeD open interval vals, the reeult intervala plus the remarb In B" is contained in &a cloeed &n1 in closed interval of the 1&1I\8 u.me volume and tba, that any cloeed closed interval in .. B" ia is contained in an open interval of twice the volume as the closed interval, (namely the open interval having bavinl the same DIne center .. dimen8ioDB thOle those of the closed cloeed interval tim. 21/.) prove (1) for open with dimenlioDa intervals. Thus the proof of (1) is i8 complete. Parts (2) and (3) follow foUo" immediately from (1). To prove (4), first note that we have vol (A - B) (A n B) - 0. 0, by (2). Define functions 11, 1a,1II,•./,: B" -+ - R by setting vol (A" /.: Eaettilll

/,(z) 11(~) -

{~

if if:c~EB E B if ~eB if:cfl B

/.(z) I.(~) -

{~

if~EA-B ifsEA-B if:ceA-B ifzElA-B

/.(z) - {~ I.(~)

ifzEAf"\B if~EAnB if z fl A (\ B. if~eAnB.

12.

BXl8T&NCB DISTIlHCB

THZ INTIIOItAL INTBORAL or THB

117 D7

L_.l .=. vol (A - B) = 0,

1•.

Then f•• /& fl" f .... .. f.1 f.l -= - vol (B), f .. l. - /.._.1

vol

= 0, and

/'.f.... f.(x) ... 1 if xxE f../a ... fAfU fAI"\B 1 ...== vol (A II B) ... O. lte \te have/l(x) have/lex) + hex) E BV A and flex) f.(x) ... A} =- f f•.•.(faUl + /.)f.) ...... fa(x) + /.(x) = 0 if x fl. fl B V A, so that vol (B V A}!"" f.. f.(x) ... 1 if x E B - A and /1(x) fa(x) f.. /l/a + f.. f.. f•f. ..- vol (B). Also fl(x) fa(x) - /.(x) f.(x) .. (8 - A) ... f... III.(x) - 0 if x f1. e 8B - A, so that vol (B = f•. f." (/&(fl - /.)fa) =... f."faf.. suppose that f.. f/1 - vol (B), which completes the proof of (4). For (5), auppose 80

80

l ..

A is contained in the closed interval 1 of E" E- and auppose suppose that IJ(x) Al If(x) I < ltI for each zx E A. Define]: setting](z) --/(z) A,]{z) :=' := 0 DefineJ: E" .... - R by settingJ(x) lex) if z~ E A,J(x) if zx f! A. For any • > 0, paR part (1) tella te1la ua UI that there exist. exilta a step ltep function ,: 1l-R ..... R luch g(z) ~ 1 for x z E A, and such that g(z) ,(z) ~ 0 for all zx E 1, ,(x)

e

Mg(x) sJ(x) s Mg(x) forall x E 1; (Mg - (-JIg» (-JIg»"" J"f,a < e.•. Then --M,(x) S Mg(x)forallx Ii f, (M, .... 2M J" f, a< 2Me followa. follows. This being true for each e > 0, I, /, J exists.

Since

~ 1,1 Ma, J,f, (-Mg) (-M,) s /,J s~ I,/, M"

we have

0

aince this is true for all of.e > (, we have since

f,J, fA / .. 1,1,

II, 1\'\ Ss MMI, ,a< Me,

and

/,1'" o. But fA I/ iais by definition 1,1'"

so 80 f ... 0, finishing (5). To prove (6), we may auppose suppose that 8 C 1, I, where 1 is 801ne Era-I. Given allY auy E> E> 0, by uniform consome closed interval in E"-I. number a> 0 auch tinuity we can find a nUluber such that I/(p) - J(q) f(q) II < E whenever p, q E 8 are such that d(p, d{p, q) < a. Choose a partition of 1I of width leaa less than 6/v"n"=l. a/~. Let this partition of I subdivide I into the closed subintervale 11, ... ••• , IN, 80 so that 1a, intervals 11, •.. ... ,,IN IN are closed subintervale subintervals of 1 whose = 1a ...· V aides are all 1888 sides less than a/v"n"=l, a/~, 1 = 11 V V·· U IN, and vol (I) (1) = =

a.

N

E vol (Ii). (II)' 1: 1-1 1-'

If P, p, q f E II Ii II ~ 8 then d(p, q)

< a,

so that If(p) /(g) I < •. I/{p) --/(q) t.

nonempty, the graph of the reatriction Hence if 1111 Ii" 8 is nonenlpty, restriction of I/ to 1;11 1; r\ 8, S, . that is,

.•. , z,,) x.) E E" : (Zl, (Xa, •.. , Z,,_I) x__a) E 1; II __ a) = z.1 t«Xa, (Zl, ••• (\ 8, /(XI, /(Xl, .•. ... , x Z..-I) x,,) J

is contained in the set

ml and M minimum and the maximum where mj AI ij are respectively the luininlulll maxinluill values valuet5 I on I; f'\ S, and the latter set attained by J Ii" Bet is a dosed l~losed interval in Jo;. }fJ" of volume {M (M;j -- 1tl;)vol (Ii) (Ij), HClwe volunle nlj)vol (I HClu~e the gl'&ph gl"aph of J / ia is contained j ) < E vol (Ii). number of closed E" the aum in the union of a finite nUlnber (~l08ed intervala intervals in ill En sunt of whose N

volumes is ia at nlost moat I: j) volulnes :E Ef vol (I (1 j) ;-1 i-I

number, (6) is iOlplied implied by (1). nunlber,

= E vol (I). (1). Sim'c Sin
Proposition. 1/ oj E" E- aucA such ",...,.,don. II A, B are aub8et1 IUb8dI 01 J: on A t.md OM em on B, II&en IAen I: AU A V B ..... - R ia integrable if&le(/rable em

Lv. l - LI+

tAot vol (A'" (A f) B) .. U&at - 0 _aM

1.1.. 1.1.

To prove thill, /.: E't -. R by thie, define /1, II, /1, I., la: E" -+ I.{) {/(:e) "() {fez) 0

l J :t liS

if:e if ~E A if :t fi A if:e(lA

#. ( ) _ { ' (:t) if I.(s) {!(OS) if:e2: E B 0 if :t ft. B S (l

,It J;

air

{/(S) if sEA'" "() - {/(z) x E A (\ B la(te) 00 if z ft. A " B. x.. if:e (l A '" B.

I'

1.. IA 1... I. have 1 ..1./ - /AfVlII.(z) --I.(z) U B IAf'\BI - O. Since Il(z) II(s) + I.(:e) I.(s) --/(z) I(s) if :esEA E AU

Then I.. /t 1. - I. I· The exiatence existence of the last lut two int. inteIs - IA II and grals gra1a impU. that I is ia bounded, 10 80 by (5) of the previous proposition we 1 -

and !l(Z) fl. A V II(s) +!I(Z) I.(:e) -!.(z) - I.(:e) -=- 0 if :es E U B, we can therefore compute

...1 L 1+ 1.1. Lv. I.. I.. VI +I. -I.) - la) .. - 1 ...1+ 1.·1. - 1/..1 .. /a - L

Lv. lI -

(/1

1 -

1

In the special case 1I=: 1, the proposition aays says that if A and B are apecial cue aublets E" with volume whose intersection has volume 181'0, 1810, then subsets of Eintenection baa vol (A V B) .. II: vol (A)

+ vol (B).

exiatence theorem. We now have the main existence Theorem. Let A C Eft be a Bet aet VJitA tDitA I/olume valums and let I: J: A -+ ..... R be a(J bount.U4junction that is i3 continuous except on a (J subaet subset oJ oj A oJ oj volume uro. ,ero. bound4ld luneticm tMt

Tiaen TAm

e:eiaU. IA I emu.

fint prove this apecial cue Let us first thi8 in the 8pecial case where A ia ie a cloeed interval 1 ia continuoUi continuous on 1. Here the proof ia and I i. is an eaay easy modi&cation modification 01 of the earlier proof for" for ft -- 1. Let MER be luch such that I/(s) I/(~) ISM lor for all aU :e s E 1. us a a > 0 eucb Given e > 0, the uniform continuity of IJ on 1 giva gives UI luch that 1/(:e) -Ie,,) - 1M I < e whenever :e, 1/(%) z, 11 Eland des, d(z, 1/) y) < a. Chooee Choose a partition lese than 3/ al Vn. of 1 of width lees V'ft. Suppose this partition eubdivid. subdivides 1 into cloeed 8ubintervals1 eubintervals11,1, ••• ,1.,80 ,1",80 that 1== lit and no two the closed 1 == 11 V·,·· V ·0·· U 1. Ila overlap except poeeibly of the 1/8 poesibly at extremities. Thus vol (II'" (llf) 1.) III) .. == • .. ,,N k. If :e, o if i,j, Iek :Ill.. 1, •.. N and j pi '" Ic. x, 11 E 1/ Ii then d(s,1/) d(s, ,,) "< 0< a, ., 80 we have 1/(:e) - J(y) It,!.: 1-+ R by -UIlI I/(z) ICY) I < e. We define II,fl: aettilll

a

e

,2 t 2..

/l(X) = == {min '_IM(Y) IJ(y) Ia(z) -M f,() {maX '/(y) J( ) II (y) •t zZ co M Then fl, It, fl It are step

{max

.xJ1I'I'BJI~ TIIII ........... L ~aTBNCS 01' or TIm INftG1U.L

119

e

if z E Ij, Iii sZ e 1" I. for any IeA: ,. ~jj aeta II, ..• ,,1. if sz is in at least two of the leta I" ... I" :Y E IjJ if s E h s e I. for any A: ~j y 1j I z 1j, Z fl. 1" k ". j ••• ,1 , 1". if zx is in at least two of the leta aeta 1It, ..... •. /1(Z) S f(s) J(z) S f.(s) /.(%) for each sz Eland functions on 1, I, fl(s) E I and lj} : Y E IjJ

• (Ii) - • vol (l). J, (J.(JI - II)It) ... EE• k (Js(JI -- fl)I.) SEe ~ E e vol (II) (1). 8inee Binee auch IUCh 1.,/. 11,1. c10eed interval exist for each > 0, our criterion for integrability on a closed impliee the existence of J, f· epeeial cue. I· This proves provee the special N

N

I-I i-I E f

I-I i-I

Now consider the general cue of the theorem, with A C .. B- a let with volume and I/ a hounded real-valued function on A that is it continuoUi continuous except on a 8ubset is not 8UbaCt of volume zero. If 8 C A is the sublet subset where I iI continuou8, haa volume aeta continuous, then A - 8 S has volunle (by part (4) of the propoeition OR seta 0 (by part (5) of of volume lero) zero) and 01 the aame same propoeition). If II we

f.1 1.1can prove that f._a can fA-. If exiete, exists, the precediJlI preceding propoeition will imply that IA I"' I -'/A-. /.1-- fA-.I. IA-af. We may therefore replace A by A - 8, L /A-. 1+ 1+ 1.1

neceaaary, to obtain the simplilyilll continUOUl on A. if necessary, aimplifyilll UIUJIlption usumptiOD that Ilis is conunuOUI intetval 1 C" C E" such definition of I Fix a closed intetvail IUch that A C 1. Extend the de6nition to 1, redefining I on 1 - A if necaary, I{s) - 0 if s E E 1 - A. nece88&l'Y, 80 10 that I(~) show that /, f exiete. exists. Suppoee 8uppoee tha~ that I/(s) 1/($)1ISM ~ M for all We then have to 8how R be defined by ,(:I:) - 1 if % E A,,{s) A, ,(s) - Oils 0 if s E 1 - A. sZ E 1. Let,: Letg: 1-. I -R byg{z) zE E 1Then /, g existe, exists, this being jUit just vol (A). Suppoee 8uppoee • > 0 Ie We can II liven. W. then find a partition of 1 such that any two Riemann 1U1Dl10r, cornepondlauch lurne for, COrnlpondt. Suppoee that this tbiI partition subeubing to this partition differ by lea Iell than e. •.. ,1.,80 V I. dividee 1 divides I into the closed subintervale subintervals 11, I., ... , 111,10 that I-I, I - I, V U ••• • •• U IN and no two of the subinterval. subintervale 1 11J overlap, except pcaibly at. extremities. poIIibly at extremiti•. 11 are a set let 01 The points of 1I that are contained in more than one II of volume zero. Let P he the nunlber 11, ... ••• ,,1. IN that are entirely lero. number of 8ubintervals lubintervale 11, aubinterYalI that have 01 these subinterYalI contained in A, and let Q be the number of suppoee 11, points in common with A. We may 8Uppoee I" ... ,1.80 ,1" 10 numbered that. that 1Iij C A if 1 SiS ~ j ~ P, that 11 I J contaiDl contains both pointe points 01 of A and pointe pointa oil of I - A if P < <jj S C 1<jj S N. Then t.wo ~ Q and that 11 Ii C 1 - A if Q < two Rieman08UIDI Riemann IIUm8

1,1

I,

,.r

correeponding to the given for ,g corresponding liven partition 01 1 are

o CI

0CI

;-1 j-t

j-P+l 1-"+1

(1;). Therefore E L E vol (Ii)'

I:

vol (11) and

I-l 1-1

vol (II) (Ii)

< e. The restriction reetriction of If to each cloeed clOlled

/,,1 mati 1,/

II, ... ,, I,. il continuous, continuoUl, 80 80 that exists lor subinterval 11•.•• I p is for j - 1, ••• .•. , P. == 1, ... , P P there are step functioDl functions laid': /1/,/": II-R II-tR Therefore for each ij = such N(x) S ~ lex) ~ N(z) N(x) for all zx E 11 II and /" (N Buch that N(z) fez) S (J' -!a') -/a') < elN. faahion: Now define a pair of functions la, /1, f.: /2: 1[--. R in the foUowilll followilll fashion:

1"

UsE 11 for lOme If sEll IOIJle unique j - 1, ... ••. , N we let /1(S) =t JiC.z) I.(s) all - /l/(s) N{s) and /t(z) I.(s) -/i(s) /1(S) I.(s) .. - -M and !I(Z)" I.(s) - M II(s) I.(s) -/,(z) -/'(s) - 0

== 1, ~J ••. • • • ,, P if j ... p == P if j ... p + 1, ••• , Q ifij --= Q Q+ if + 1, ... ,,N. N.

U z El, for more than onej .. 1, ... •.. ,,N~aet N ~e Bet IfsEI/formoretbanonej -1, /a(Z) I.(s) - -M

and II(s) I.(s) - M.

Then /a,/. /I(Z) for all s E 1. 1.,/. are atep functions functiOl'l8 on 1 and /1(S) I.{s) S /(s) I(s) S I.(s) Furthermore, making a repeated applieationof application of the preceding proposition,

" 1"J" (f. -/~ - !t 1: I,J, (f, I-l

~

~

- I; 1" /" I-l ,.

~

- 1-1 L 1" 1: /" I-l

J::.

(f. -Ii) (f, -/~ (f, -/~ + (f. -Ii)

00

I; /" +"'1'+1 1-1'+'

(fsl-li) (fal-/i)

•o

"N

-Iv + 1000+' (f, -/~ I; J" (f. 1-0+1·

(f. -Is)

0 0

N

+1-... L+a /,,2M f,,2M +1-0+1 /,,00 1: I; J" 1-4+1 1-1'+' "

~ L vol (11) (Ii) S •I + 2M. 2M, - .(1 + 2M). S 1;:. + 2M I; l"I rf

/-1'+1 l-JI+l

Since I• WAI wu an arbitrary positive number, our criterion for intecrability on a c10Ied exists. The proof is ia now DOW comcbed interval again apin implies impli. that /, I exiata. plete.

f, /

80 far the only subsets eubleta of ESo E" that are known to have volume are closed intervale, seta eete of volume aero, intervals, sero, and eete sets that may be obtained from theae these by put (4:) (4) of the proposition propoeition on leta of volume aero part zero (page (pap 22&). 225). The theorem ...,.. ua to live pve other examples of leta . . . . 111 sets with volume. Exuaor.L Let 1 be a closed interval in 8-B-11and let 'PI, ExAMpu. 9>1, fII f/Js be continucQntinuoua real-valued functioDl on I such that 'PI(s) OUi functions 1 9>1(%) S ~ fII(S) f'I(~) for all sz E I. Then the_ the let (s., •••, "', Sa) E Era E- : (ZI, (%., ... z..-i) E I, 'PI(%., f (ZI, •••,, %a-I) ~(%1, ... , %11-.) %a-l) < Sa %. < fII(%., ~(%1, ... ••. , %a-') %.-1»)t hu volume. For if MeR MER is such that I9>'(z) fI.(%) I, I1\0.(%) fII(%) I :S M for all % hal zE 1

B" iI it the closed clOlled interval and J C B-

s.) E S11-:: (Za, (St, ••. z..-i) E I, s. [-M, I1<.., (Zl, ..• • •• ,, z.) ••• ,, z.-a) z. E [M, Mlt, M) I, tbeD baa the value one at each point of the theD the fuctioa function I: /: J -+ - B R which hu

qu.tion and value IeJ'O . t in question aero at all other pointe points of J is not continuoua continuous onl7 at pointe pointa of the form (s., z..-1» onI, (ZI, •••• • • , s_., %.-1, fI.(S., 9'1(%1, ••• • • · , :l: (Z., ••• • • • , S~l' ~'m.l, a-l» or (%It .,.<St, •.. ,z..-i). th.-e latter are a set of volume zero, ..(Ss••••, z.-~). Since Bince tbeee zero. foIl exiata. exists. Wuetrateci in Figure 36. Thie •Ia WUltrated

/,,1

•

18. Jft&ATD IImIGIW.e

JSl

F1ou.. 36. The wdt baIlla JlI hu YoIum•• Except for .... of YoIume HrO It 18 ....do wlcbed bet_ the anpba of the oontlauoua real-Yaluecl fuactiou • .,:lP-a• •ha'e .,(~, ill) - ..; 1- al'- ..' If al'+ ..' S I aIld .,(~ ..) - 0 If ~,+ #Ia' > 1.

IS. ITERATED JlIlTEGRALS.

When the integral of & continuous function of one variable actually baa to be computed one usually U8EI8 antiderivatives. The main method for computing integrals of functions of several variables is reduction to the one-variable cue by means of iterated iotegra1s. In the following. for any sets ACE- and BeE- we identify A X B with & .ublet of E-+- in the obvious way: A X B - (%1, ... , %.,1/1, .... fl.) E E-+-: (ZI, "'1 %.) E A, (1/1, ..., 1/.) E BI·

Let ACE-, BeE- mad let I: A X B -. R. Suppoae /AXll I 1M lunction 1(11): B -. R given by 1(11)(1/) =I(z, 1/) U ~ on B. TAm il 1M lunction on A whoae lHJl~ at ead& :e i, U ~ by 1taH TluJorem.

m.u and tAat lor eacA % E A

/./(11)

/./w

LXJl/- /..(/./). (2'Au ~ if 011- wriUM ita 1M .u,Atlr IRON IraftIpcIrent IO'I"M

LXJl/- L(/./(%,1/)d1/)tk.) If we extend I to a function 00 E-+- by defining it to be zero outside A X B, the theorem becomes equivalent to the analogous statement for

_232

X. x.

IIVLTlPLII INTEGRALS IHTZORAL8 IWLTlPLIJ

the special case A = == Eft, g.., B == = E"'. E .... In this special case, the integrability of I/ on E-+sublet of B-+- inlplies implies that I/ Jllust must be zero outside some bounded subset P+-, E"+-, 1 and J being B-+-, therefore outside 80nle some closed interval 1 X J of g..+-, cloeed i. closed intervale intervals in Era E" and E'" E- respectively. It follows that the theorem iB equivalent to the analogous statement for the restriction of / to 1 X J. Thus At B are closed cloeed ThUi without any loss lOll of generality we may assume that A, intervals E", E" II- respectively. We make this 888unlption, assumption, and first prove intervala in Era, the theorem when I/ is a step function on A X B. In this case, cue, for any

f. 1.1c.)

~ E A the function fl.) /(.) is a step function 011 B, 80 that 1<.) exists without any further 8S8umption. assumption. If fl' /1, fl' /1, ... , f. /. are step functions on A X BBand and • the theorem holds for each Ii, fit then the theorem holds for E Ii, fj, for 1-1

( t/I- t 1

jA)(JI I_I

1-1

A)(JI

11-

t 1(rJ. II) -1 (t 1.( II) -1 (rJ. til)' I_I

A

A

I-l

A

1_'

But any step function funetion on A X B is the lum Bum of a finite number of step functions on A X B each of which is of the following aimple limple type: there are subsets 8 11,, ••• t, 88"....", H, each 8, •• of R, 8. heinl being either a lingle point or an open interval, such that {(ZI, f (~I, ••• .•• , :r,.+",) :.:,,+_)

%1 E 8 B1,1, .•. ••• ,, ~.+z..... E 8 B,.....) E Era+8"+- : :.:. ... )

lubset of AX B and the step ltep function baa is a subset has a constant value c E E R on thil subset IUbaet of AX B and the value aero aubaet. this zero on the complement of this thileubeet. For a step function on A X B of the above simple type the theorem can aide of the equality in queation be verified directly, each side question reducing reducilll immediately to c times the product of the lengthl lengths of the leta 8 1, ••• , 8 ...... .... Bence ltep function on A X 11. Suppoae, the theorem holds whenever I/ is a step Suppose, iI any function satisfying aatisfying the hypotheses of the theorem. For finally, that / is any eI > 0 there are step functions II, fl, lion It on A X B such IUch that /1(') /,(,) S /(.) ~ S ~ 1(') 1.(,) (f. --/~ II ft(') for all , E A X B and (Jt Ii) < e. We then have fl S :S

fAXIII S JA)(1IJ:S

fAx.I•. fAX.ft.

1(.,(11) S (Jt) (f,)(.,(y) fl.) (1/) :S (.) (y)

But

f. Itfl and f.

lAX. fAX.

fAx. fAX.

~ E A we have (f~(8)W) S (Ji)(8)W):S so that (f~(., (f~(8). for each 1/11 E B, 110 (Ji) (.) S :S f(.) S :S (Ju(8). I. are step functions on A and It Furthermore, for each Furthennore,

I" ftI. can be found for any. Since such fit can therefore write We ran

J.

f./(., f. f.

> 0, it follows that

L (/. I)f) exists. L(/. existe.

'3. 13.

ITBBAftD INftGIIA.IA INftGB.UA IftIlA.,-aD

lSI 131

or

[XlIII Lxa /l S~ [L (f. (1./)I) S~ [XIII.. Lxa /.. Combining this with Lx./1 S ~ IAxal fA XII I S~ LXII Lxa /• /I we get ILx. l - L(f. I) IS ~ Lx. CI. CI, -111 -/~ < .. (1.1) The theorem follows fronl fact that the lut Jut inequality inequa1i', holda for aD)' AD)' from the faet f > o. .>0.

.

We clearly have the 8yulnletric reeult that if I is ia integrable intecrable OIl on A X B symmetric result then

"s,

fAx. 1- I.f. (fA fA I(s, ,)cfs r)ds exiata exiIta for each , lAx. (IA I). I), provided that that. IA

e B.

Corollary 1. integrable reaHtaluecl/UfldiM& retJl...wJlu«l/uRCtiarl, on Oft 1M ..... A oJ 0/ I. Let I be tlR an i7&t8grable E-, let B be (Ja clo,td E-, cmd G1Id let T...A: X B ... be eAt ",. ... cloatrd intmHll interval in B-, A: A X - A "1M ~ on factor, IJuJt W"A(%, ,) 1/) -• sz if sEAt ,E II E B. 2"_ on. the first firMlactor, that ii, 1I'A(S, i,f sEA,

7'_

fAx./otrA"- (fAI) IAxa/oTA (IAI) vol (B). (In tmotAer anot1&er not4tion, notation,

BO 80

fAx./(s) dsdr ( [ I(s)cfs)(f. I(s)ds)(!. .).) dr).) IAx./(s) cfsd, - (L

For z E A and ,E 11 E B we have CI (J 0 'rA)(.)(') (J. TA)(S, 'r.)(., ,) ,) -/(a) TA)C-'(') - CI. - I./(z) -/(z) vol (B). If IOTA that I. CI 0 TA)(., trA)I.) .. /. trA is ia batepable iatepable _ GIl

f.

f./(s) -/(s)

(L (/..1)

1.

A X B we get IAxa fAXIJ I/o0 tr1I'Avol (B» .. I) vol (B). lienee we A - L (J (jvol A exiIta. exiata. To do tbiI~ we" nduoe tID need only show that Lx./o0 1I' trA to the case A Era by ex'tendinll to • function on Bra tha, _ _ \1ft .-0 M cue Eextending I a fullCtion OIl II- thM ........ ...., M eaeh point of E" - A. If we then ehooee chooee a cloeed intenall each interYa11 C ..... II-IUCh thM that Ii. eaeh point of E-1, we reduce to the cue - I. ThU", lis lero at each E" -I, caM A -I. That.iI, we may . .ume that A is a closed interval in P. nlay _urne Be. ThiI beil'llO, the ....... exiateDoe of IA II implies that, given liven any • > 0, we 0Ul Incl Itep atep ··f\metionI/-./. f\mot.iau 1-./. _ ClUl 8Dd GIl A

Lx.I

-Dl.,

fA

such that 11(s) II(z) S I(z) I(z) S I,(z) I.(s) lor each -.eh az e EA. . /A CI. luch anti (I. -IS> -/~ < .. ".. A, I. Aare step .tap functions on A X B aucb such that ~ ,.. for each eMIl.• E A X B (f, 0 1r'A)(') 1I'A)(') S~ CI (f 0 TA)C,) S Clio (f, 0 TA)(') and 0 TA we have Cli trA -I. OTA)0 trA)"

fA

II

11' T /1 0 TA,/. 0 W"A

fAx. -I.)

(L (1.

trA)(')

lAX. (J, (f. -I,) 0o1l'A (f, 1r'A Cli exists, as was to be shown.

-IS» -/~ )

trA)(')

vol (B)

L_C/. L_u.

n-

Lx.-!.

< • vol (8). (.8). B81U!8 Hence /..... !. TA trA

If we apply Corollary 1 to the cue 1/ - 1, we pt the followia& followilll aimpl. limple ACE- hu B- is a eIoIecI result: if ACE" bu volume and Be BeEcIoeed iD&ervai, ira&erval. &hen then vol (A X B) ... == vol (A) vol (B). In particular, if A hal RIO IeIO volume 10 baa _ AXB.

1\ iI worth remarkiDl at this point that the theorem remains true if we n .. remarki._t replace the UlUlDption UIUIIlption that 1<-) /Ca) iI is integrable on B for all Zz E A by the .-umption 4. - 8, S, where 8S is a subset MIUIIlption that/(.) that/Ca, is integrable on B for all z~ E 4 01 eleof A 01 of volume valume 181'0, if we then understand underatand /./(8) to be an arbitrary elament S. To see lee this, note fint IDtIlt of lOme eome bounded sublet eubeet of R whenever s E 8. t.ha' X B) -.. O. o. Therefore that B may be UlUmed IllUmed bounded, 10 that vol (8 X I~/- O. If we define,: A X B-R by,(.) -/(.) if, E I~I - 1(') if. E 8 X B, other-

1.1(.,

-

lAx.' IAXB'

lAXIAXB

IAx.l. !AXB/.

wille -0. Therefore U wile ,(.) - 0, then (f --,) g) = But CJ 88 I/ to (A - 8) S) X B and is (f - ,): A X B -+ - R baa the 8&IJ18 same restriction as 181'0 on 8 X B, 10 that lA-~xJII exists exista and equala (f -- ,) ISO equals U fI) I· ThUl ThUi

LXB Lx-

.lA-.,XB /

LXB Lx. /.

LxJl l - l~xJll- L... (f./) - L-. (f. /) + f. (f./) - L(1. 1). CorollGry ., A C B-1 E--I be ~ COfII1HId and haN POl""" volume IIU ", Corollary J. t. Lel 1M 1M tile .. and ,., fPl, /vndioftl ad tAat IIuJt 9'1(S) 411(2:) S fPI(z) f>I(z) lor lor all z:c E EA. 9'1, ".: fPI: A ..... - R be COftIinuoua coMnUOU /ad ... IUCI& A. coMnUOU retJl...tIalued retIHtaluMI /urtclitIA TAM if I/ ... g tI COftIiftuoua /uftdion on 1M tIN .. ad

2'_

Za-u

((Sl, ••• ... , Za) z.) E E(Sa, ... 8B - I(SI, Era : (ZI, ••• , z..-a) E A, 9'I(s.. • •• , s_a) S z. S 9'1(3:1, 9'l(Sl, •••• s...a> :S \OI(Zl,

••• • • • , s_a) s...t> J, ),

.. ,... Ii - IA <[.'1), ... r.../ .. 1M fat:timt on, A - - IIfIluc at

(

- -\ . L..

. , S" "',"'-11 U

t-a·....a.-a''f(S ..... ,~ - \.01.. ..,PI..........,

Let B B be aa closed cloeecl interval in R containing 9'I(A) t'l(A) U fPlCA), ~(A), 10 that 8 C A X B. Extend J / to a_ function on A X B by Betti. B aettinl 1(') /(.) - 0 if •e A X B - 8. Then I/ itia bounded and isia oontinuoua continuous at any point of is Dot not of the form A X B that iI

•e

(s..... , z..-I, 9'I(s., ••• , z..-a»

or (ZI, ••• , Sa_I, IPt(ZI, •.• , z..-a».

'l"I-.later form a let 01 of volume 181'0, TheIl later point. points fonn zero, by part (6) of the proposition Since A X B baa volume (by the comment Binee COll\nlent following CoraDarr 1), exiata. A1IIo AlIo for each (ZI, CoroI1ary exists. (~l, .•• ••• , Za-a) ~.-a> EE A,

OR . . of volume 181'0. OD leta sero.

1.1- IAxJl IAXBIl

1 ............. ... ... ,, z.)'" ...PI , /(SI, .... •••• .-.-.> r I/....... .a.-a)' It 1( "(Sl•.•. Za)=. - /. /.•• <8 .......... I(~I, ••. ••• ,t z.) Sa)dz•• J. . ~.(8a ......__a) J. ... (II,...... HeDGe the theorem is ia applicable to the present A, BeD. A B and I. I, giving 118 us the

-'·'l· J

..... rt!8Ult. n.ult. ......

,

'4.

CBAHOII OJ' VA.IIUBLII

J35

In favorable cireumatanCM circumatances Corollary 2 may be applied repeatedly to express an integral over a sublet subset of EE" &8 an n-fold iterated integral. If we apply Corollary 2 to the cue case where" where n ... 2, /I -= ... 1 and A ... == [a, b) BOrne a, H, aa < b) we obtain the well-known fact that the area of (for lOme tI, b E R, the plane set bounded by the lioes s .. y ... 1P1(:Il) ~(%) lines:ll~ - II a and and:ll - b and the curves 1/ (",,(:Il) - 1P1(:Il»d:ll. Howev~r we have actually proved and 11 71 - 1Pt(:Il) ",,(z) is (",,(z) 'Pl(Z»•• Howev~r 80mething BOmething nontrivial because this is now a theorem, not a definition. SimiI... 1, for any compact subset A of the plane that has larly, if n - 3 and I" area, the volume of the subset of ga lying over A and between z = IJIlying == 1Pl{:Il, 'P1{z, y) :II:

ID

f

and .... ",,(z. 1/) is z .. ",,(:Il,

fA (",,(Z, (1Pt(:Il, 1/) -

'Pl(Z. 1/». 1P1{:Il, 1/»d:ll d1/. dy.

, •• CHANGE OF VARIABLE. •••

Lemma. Let D be II G com_ aubI,t U 01 B-1 TAm there comptICC aubNt eubNt 01 oJ the open lUlnet oJ E". m.t IUb,et1 D', V of e:z;iat ",buta oJ U, 'With with D' comptICC and V open, auch that

com_

DC VCD'C U. DCVCD'CU. Buch that the closed Each point of D is contained in an open ball of Eft Era such ball with the same center oenter and radius is entirely contained in U. Since D is compact, we may find a finite set of such open balls whoae whose union contains D. We may then take V to be the union of this finite set of open balls, D' DI the union of the corresponding closed balil. balls. PropoaitioR Unity"). Let D be a compact aubaet 0/ oj BPropoaition. ("Partidon. ("Partition. oJ oj URityU). E" and l8t (U,I (U.I.Es oj open iub8eta E- whoae whose union contaim containa D. Bubaet.t 01 0/ E" ,EB be a collection oJ TAm tMre there ill i8 a finite set Nt 01 0/ continuoua continuous Junctiona t/lN: E" juncticma t/I., ... , 1J!N: E- -. [0,1] [0, 1] auch that

"'1, ... ,

"'1(:Il) ~1{S)

+ ... ··· + "'N(:Il) ~N(%) -

1

lor taM sED taM "', !/Ii i, compact sublet BUbNt 01 Bets Jor ttu:h :Il E D and eada i. aero uro oulaids outBids a(J comptICC oJ one 01 oJ tAf ~ Bell ( U,I ,EB• (U.I.ES. Buch that 1(x) h{x) ... Start with any continuous function h: A: R -. -+ R such == 0 if h{:Il) > 0 if x > OJ for example, we may take 1&(z) h(x) == = zx for z:Il S 0, while h(z) z:Il > O. Then the function ,: R --+ R given by ,(z) ,(:Il) .. 1(1 1&(1 - zI) has the propertiee ,~, < 1. Hence if properties that g(z) ,(:Il) - 0 if Izi 1:Ili ~ 1 while g(z) ,(x) > 0 if 1:Ili " > 0, then ,(n) is sero aero for Izl 1:Ili ~ 1/" and positive for Izi 1:Ili < 1/". For each P E D choose ohoose '. fl. > 0 such that the closed cloaed ball in E" of center pP and point p U,I ,E •• Let B Bpp be the radius 1/". is entirely contained in one of the sets It U.I.Es. open ball in Ea p and radius 1/" 1/11,. E" of center P .. Since D is compact there is = BpI B.I V .... V B.NI B pN • a finite subset PI, ... , PN of D such that DC D C U == ·· U I

•

"'

_ 136

X. IIVL'ftPLII INTIIORALB X. 1f171JftPLJ: INftORAL8

Set , (II" d(z, Pi»~ N

I: g (I'JIlI d(z, 'Pi» i-I

for,; ia a& continuoualunction continuous lunction on the for z E U and ii-I, - I , ... 1'1', N. Then each tp4 fIJi is open aet set U::> D with values in [0, 1] and 4PI(%) CPl(Z) + ... · + tpN(Z) fPN(:a:) - 1 for all % E U. In addition, for each i = I, ... , N, the points where tp4 ia not aero = 1, qJi is zero are contained in a compact subset of one of the sets I U.}.ES. U.I.e". Now use the lemma to obtain subsets D' and V of E", respectively compact and open, such that I

I

DC VeD'C U. DCVCD'CU. Apply what has haa been proved above to the compact aet set D' and the collection of open sets I(V, V, U - DI D) (whose union contains 0'). D'). We get functions analogous to the above tpla "/8 and these we group into two batches, according is not aero lero is conto whether or not the set of points at which the function ia tained in a compact subset aubset of V, then we add the separate aeparate batches. We get continuous functions lunctiona 'I, '. on an open set U'::> 0' values in [0, 1] D' with valUeI luch that '.(s) ,,(%) + '.(s) 'I(Z) ,. 'I is aero zero at each point of auch - 1 for each z E E U' while '. IUbaet of V and '. 'I il U' outside outtide a compact lubset ia aero zero at each point of U' outside outaide a& compact sublet aublet of U - D. Since Bince 'I(Z) - 0 for each z E E U' u' - V and ainee '1(';) - 0 if Z D'.. Alao'I(%) Also '.(z) - 0 if zED, Bince V C 0', D', we have '1(2:) 2: E U' U' - D' 80 that '1(%) 'I(Z) - 1 if zED. % ED. For i-I, ... , N N.. we define "'.: t/I,: Eso E" ..... - R by t/I(%) .. V' and 1/Ii(%) %E E" - U ('\ ",.(z) - ".(Z)'I(%) tp~Z)'I(S) if % Z E U ("\ ~ U' "'i(Z) .. 0 if sEE" ~ U'. Since Bince "', i. sets U ('\ ~ U' and E" Era - D' (it is aero IeI'O on the "'. ia continuous continuoua on the open seta latter), it is continuous on their union, which is Era. The other desired propia E". erties of '/II, ••. ,1/IN follow immediately from their construction.

+

'1,"

II:

"'I, ... ,"'N

Corollary. Let Let, D be II sublet 01 oj the 1M open open, IUbad 0/ E". Era. TAm a compaet aubaet aubaet U 01 Tlaen Junction "': 1/1: E" ---+ [0,11 (0, 1] BUNl tAm ",(,;) ~(%) - 1 1M' there is 40 continuoua lunction 8'IA!1& lAst lor eacA :r; E D tmd ond "'(x) ~(x) = 0 lor Jor each % outBith some Btnne compaet compact aubBd 0/ U. zED x 0tAt8itU aubaet 01 u. Thia This baa has eaaentially essentially appeared in the proof above, but it also follows easily from the atatement statement of the proposition. Let 0' D' and V be aubaeta sublets of Era, ,respectively compact and open, luch such that E", DC VCD'C U. DCVCD'CU.

Bince 0' VV Since D' C V U (U - D), we can apply the proposition to the compact let 0' D' and the collection of open seta sets IIV, D'} to get continuous continuoUl funcV, U - Dt tl, ",.: ~t: E1/11(Z) ",.(x) tJll(Z) -== 1 for each zED' :I: E D' and tions "'., tiona E" ..... - [0, 1] such that "'1(Z) ~1 and "'. tS'1 are sero zero outside compact subsets sublets of V and U - D respectively. Since ~l is 1 on D and aero lero outside D' , we may take '" # -- ~. #to outBide 0',

"'I

+

"'I

,4.14.

CHANG. 01' or VAIIUJILII VAm......

=

137

"'1I, ... ,"'N

We remark that the functions 1/IN of the propoeition proposition and &Iso also the function til 10 .&I. to palla ~ all partial '" of the corollary nlay may be chosen 10 derivatives of order m, for any given po8itive tn, by 1tarti01 poeitive integer m, ltanin, with h(x) ~It•••• ••••, #N, ~ poIIeII ~ h(z) - x·+ Z·+I1 for z > o. O. In fact it can be .hown that "'I u" lor all partial derivatives 11(,;) -• tr for 1/1 a > 0 (cf. (cl. derivativel of all orders orden if we take "(:c) roll" Problem 26, Chapter VI).

"'N. '"

function If on the cloaed closed inlmlall interval 1 01 E- ia i, ~ i1&legrtJble Lemma. The real-valued lunction on 1 il and only iI, if, lOT for each f > 0, there eziat exiBt continuOUI ~ real-fxJlued fundioM jundioaa /1, J. f2 on 1 BUCh II, BUCk that 11(x) II(z)

for tach z %E 1 S I(x) I(z) S II(x) I.(z) lor

and

J, (JI -!l) e. - It> < •. makes repeated Ule The proof nlakes use of the tint propoeition proposition of 12. Firat Fim suppose that J I satisfies aati8fies the given condition. Then given •e > 0 there are continuous functions fit 1-+ R 8uch such that II(z) Il(x) S fez) /(x) S /.(z) I.(x) for each Ih It: I,: 1(JI --/1) /1 and /. :ez Eland (J. I.) < ./3. e/3. Since II I•._are intepable on 1 there are ltep functions f.', N, N', /t" on 1 such that la' N(z) step f1". N, It'. /t" (x) S/I(:C) S /l(Z) SN'(z) S /,"(,;) and /t'(z) S /.(z) S N'(z) /t"(z) are true (or for each :e z E I and J,(ft" -/1') _/3, N(z) -II') < ./3.

J,

J,(N'

(/t" -!t') - N) < f/3. f,J, (N'

functions The step (unctions I(z) E I and f(x) S /t"(z) I,"(x) for each z Eland

N, N' /t" are luch that N(z) Il'(z) S ~

J,(ft" (Jl --It') J,(N' -- ft') N) = f, J, (N' -It) - It) + J, J, (J. -It) - II) + J, (JI N) S f, /1') + J, (JI -I.) Ul" -It') J, (N' - N) -/1) + J, VI" - N) < ;

+ ; + ; - ..

Thus/ Thus 1 is integrable on 1. I. To prove the converse, convene, IUPpoee IUppoee first firat that any fun('tion on 1laatisfies step function satisfies the given condition. We reuon in the laDle 1&018 way &8 above. If IJ is i8 integrable on 1 I then for any as any.• > 0 there are step functicma functiona 110 8uch that It(z) /t(z) S /(z) II, lion 1 such I(z) S II(z) I,(z) for (or each z Eland (fl (f. - It> I.J < ./3. Since step functions are 888unled I88Umed to satisfy aatisfy the given condition. f/3. condition, there are continuous functiol18 /1', /1": 1 .... such that we wUl wiD have functions 11', It', II", /t", It'd": - R lOch N(z) S /I(Z) ::; S 11"(z) N'(z) and /I'(Z) /1'(X) ~ /1(%) /t'(x) S I.(z) ft(x) S 1t"(III) I,"(z) for all IIIz E E 1 -and ·and - N) < f/3, (N' (It" --It') (N' -N) N) < e/3. Thus the continuoua functiona functions N,It" 8u('h that 11'(x) N(z) S fez) /1', /t" are 8uc'h f(x) S N'(z) f·l'(z) for all z Eland

f,J,

f,J,

J,

(It" - fa') N) = J, (N' (It" - It) -/1) + J, UI N) f,J, (N' I.) + J, V. UI -It) Ul --Nl J, (J," - N) + J,J, (JI - II) J, (JI" S f, I.) + f, (Jl" - N) fa') < -j. i· + ; + ; -

--t

e,

,_tiafi.

.. that l-tisfieI the given condition. Therefore it remains only to show 10 \hat Ulat for any step 8tep function II on 1 I and any eE > 0 there exist continuous that functiona lunctiona 1.,/. /.,1. on 1 such that /a(S) /1(%) S ~ /(s) /(z) S :s; I,(s) /.(z) for all sx Eland E I and C/e -/a) CIt -/1) < ..eo But any step function on 1I is the sum of a finite number of Itep lunctlo functioDl Rep .. on 1 each of which i. is of the following simple type: there are ..bIetI 8 t , ... ••• ,, 8. 8,. C R, ft, each B. 8, being either a single ainille point or an open IUbIetI 8., in ....., IUch that the set {(%1, (Sa, ••• , z.) B" I is a interval, :tIt) E E" : Sa %1 E E 8Ba,1, ••• , sz".. E 8,,) IUbeet baa a constant value c E R on this subset aublet, of 1 and the step ltep function haa and the value 0 on the complement of this subset. Therefore we need only prove tbat that a& step ltep function on I of the above simple sinlple type satisfies the indiI, ... , n the subset B, cated condition. 'nlat is, if for i ... === 1, Si C R is a single singl(\ poiot (Zl, ••. ••• ,, z..) ••• , point or an open interval such that I(Zl, z,,) E E" : s. %1 E B SI,I, ... ~ E B.t 8,.) is a subset of 1 I and if I: /: 1 1--+ s.. - R has the constant value c E R on this subset its complement, aubeet and the value zero on ita conlplement, we must show that for 1 --+ R such that /I(s) /1(%) S ~ any •t > 0 there are continuous functions /1, / .. /.: 1I(s) S It(z) I.(s) for lor aU ,; %E Eland (J. --I" /I) < e. IUfficea to prove 1(,;) I and (f. f. It clearly suffices this lor c - 1. Allume that 1 happens to be the closed interval in E" deterthle for (11, .•. ••• ,, a., 0., b ••• ,,b,.. Firat suppose that some 8, mined by aa, b.,1, •.. b.. First B. is a single point, say B. 8 1 .. (II E (Gl, btJ. bl ). For any a > 0 choose a continuous function poiot, - CI. E (a., fP: R 1] such that ';(CI.) c;(al) - 1 and .,,(%.) CP(%I) .. Zl - (tIl .,: - t (0, 1) - 0 if IISa CI.I > a. Define #: E1] by 1/1 (ZI, •.. , z.) ~ ~(z) t/!(z) for Cor all 41: E" .... - (0, 1) ~(s ..... z..) .. - CP(Xl). ,,(SI). Then 0 S /(%) /(s) S sEland sEI and

f,J,

J,

°

f,J, (41(~ -- 0) S~ 2a(b, 21(ba -

a,) as) ..• · · • (b. - a.) ,

taking asmall enough. It remains to conwhich can be made 1_ Ieee than e by taking' sider the ease cr, < fji fJ, S bi • .... Yor case where each B, Si = == (cri, (a" fJi), ~i), where Oi S ai ~ b.o o r any that 2a 23 < fJ. fJl - ai, a,., choose a continuous function aa> 0 such tbat cra, ... ••• , fJ. - cr", /1: Ela: E" - (0, 1) such that /1 /a is 1 on the closed interval determined by QJ I, •.• , a" fJt - a, a, ... era + a, CI" + a, a, fJ. ••. ,, fJ. fJ" - aa and 0 outside the open interval detemtined a., fJ.. fJt, •.. , fJ,., and choose a continuous function determined by ai, era, ... , CIro, • .. ,fJ", /.: E•.. , a,., I.: E" -.... (0, 1] that is 1 on the closed interval deternlined determined by aI, cra, ••• cr", /l., .•• ~It ••• , fJ. (J" and 0 outside the open interval determined deternlined by CIa al - a, ... , a,. a, fJ. fJt + a, ... , fJ" I. Then /l(Z) /.(z) for all s% Eland E I and CIro - I, I, "', /a(S) S /(z) /(s) S /.(s)

+

-/a) S~ f/.. ... (J.(f. -/1) J, (J.(f. -/1) -II)

S (fJa 24) ·••. (iJ" - cr. cr" + 2a) «(Jl - CIa crt + 21) • · (fJ" 23) -

(fJl - al cr. - 28) ... (fJ 2a). ("1 (13.... - acr".. - 28). 0

•

0

Since polynomial functions are continuous, this latter expression can be made Blade less than. by taking 8a sufficiently near zero, and this completes conlpletes the proof.

14.

vuua'"

CHANd 01' OJ' VABLULB CllANOJI

239 139

Theorem. Let A be an. 0/ E·, E", I(J: f(J: A -+ --. /:I.'" A'" 4a one-one continu41& open aubaet aub,et oj DU8ly who8e ja.cobi4n jacobian J is nowhere zero on A. Suppoae Suppose owly differentiable map m4p whOle J.,~ i, that cp(A) --+ outBide 4a co,np4Ct colnpact aubaet aub8et oj of I(J(A) tp(A) 4nd and tIuJt the Junction /: J: I(J(A) -+ R is i, zero outaide

that tIuJt

LIAl h'A) J exislB. exiata. Then

Since the proof is quite complicated it will be given in a number of steps. We first make a few preliminary remarks to be borne in mind below. The inverse function theoreul f(J(A) is an open subset of Era E- and theorem implies that I(J(A) that the map I(J-I: f(J-l: I(J(A) ~(A) -+ A is also continuously differentiable. Any comCOD1pact subset of A (or I(J(A» ~(A» is mappe4 luappecJ by I(J tp (or I(J-I) ~-1) onto a compact cOlnpact subset of I(J(A) ~(A) (or A), since the inlage image of a ~ompact ~mpact set under a continuous map is compact. Similarly Similarly,J since lince the inverse image of an open set under a confint proposition of Chapter IV), I(J " induces a tinuous map is open (by the fi1'8t one-one cOJTe8Pondence betw~n the open lIubsets lubsets of A and thOle those of I(J(A). ,,(A). corresPondence betw~n &ll8umption that exists is superfluous, for this If !I is continuous the a88umption from the &ll8umption fact follows automatically fronl assumption that 1 f is zero outside a compact subset of the open set I(J(A). CODlp&Ct \O(A). The reason 1 J is &ll8umed asaUDled to be zero compact subset of \O(A) I(J(A) is ill that one must IllUSt allow for the eventuality outside a cODlpact A,I(J(A) of A, ~(A) or JJ.,~ being unbounded. As usual, the component cOlnponent function. functions of fP t{JJ., ••• , I(J", f/J,., 80 that \O(x) ('Pl(x), ••• •.• , 1(J.(x» \O,,(x» for all I(J will be denoted by 1fJl, l(J(x) = (lfJl(x),

f."Al !,,(Al

J., - det( zx E A and J"

:= ). :=).

(1) The theorem is true if I(Js(xs, ~l(Xl, ••• ••• ,, x,,), ••• .•. , IfJ,.(Xl, f,O.(Xl, ••• ••• ,z,.) , XII) are a Xs, ••• ••• ,,x". I<'or if f(Jl(x), lfJl(x), ... ..• , I(JII(X) permutation of %1, x•. }4"or ~,,(x) are just Xl, ••• ,x., , x"' but poaaibly in a different order, then J J.,~ is the determinant of an n X n square possibly array that baa has precisely one 1 in each row and each column, with all the J., = :1. other elements zero, 80 that J. ±1. Thus the statement statenlent is a direct consequence of the definition of the integral, which does not depend on the order in which the coordinates are taken. &ll8ume that the theorem is true for n - 1 in place of n, (2) We may 888ume if ft n > 1. For suppose suppoae we prove the theorem under this &88umption. assumption. Then if we prove the theorem for n - 1 it will be true for n --= 2, lince since true for fa n -- 2 it will be true for n .. - 3, since true for n - 3 it will be true for n - 4, etc., 80 the theorem will hold for all n. I is continuous. For (3) It is sufficient to prove the theorem when 1 suppoae it is known in this special case. Then given an suppose all arbitrary I: I(J(A) cp(A) -+ R compact subset of I(J(A) which is zero outside a cODlpact ~(A) and integrable on I(J(A) cp(A) we UOI(J) IJ.,I must show that A U 0 ~) I J" I exists and is equal to J. Let DC D C I(J(A) ~(A) I is zero outside D. Apply the previous corollary be a compact set such that 1 I(J(A) to get a continuous function l/!: to D and ~(A) 1/1: E" Eft -+ (0,1) (0, 1] that is 1 on D II:

!fA

J.,IA)/. !"IA)

... . .

X. IWLTIPLIJ 1CVL'l'1P.... INTIJORALS INTIIORALI

and ~f cp(A). Let 1 he a closed interval in E" E_d 0 outside a compact mbset mbeet D' of that ,,-'(D'). For convenience, if F F is i8 any tliat contains the compact set D'V cp-I(D'). function on a subeet shall denote by F P the function on g.. Eft which suheet of E" H" we Rhan with F i. sero lera elsewhere. Thua ThUi F where the latter is defined and is

apeee apeea

J,

f,

}.(Al I" /, 1. J. Now 8uppolle J ate, }.CA) 1suppose we are given Kiven some BOrne •e > O. Since /, 1 the lemma enables us funr-tions (II, QI, (I.: (/,: 11--+ R such 8uch that ua to find continuOU8 continuous functions

J, (" J,

~1(:r;) (l1(:r;) for each % xE Eland {(I. - (II) ~ J(x) ~ I,{:r:) I and It) < t. f. Then t/t(X)(lI(:r;) ~(:r:)I.(:r:) ~ J(:r:) I and (I/Iu, - "I) !/Igt) ... = ~(g, - (It} It) ~ S l(x) S~{z)g,(:r:) St/t(X)(lI(Z) for each :r: ZE Eland (I/IU' t/t<us Cut tji. ~ respec<us - It> (It} < f. t. If U we let fl, II, f, II be the restrictions to cp(A) of iii, (l1(:r;) I.(:r:)

J,

J,

tively, then '1,/1 11,1. are continuous real-valued functions on ",(A) cp(A) which are J8l'O I,(z) for each :r; % E cp(A), and f~A)(J1 f~Al(J, It} It> I8l'O outside D',ft(:r:) D',il(:r;) S ~ I(z) I(x) ~ 11(:r;) < f.•. Now consider the real.. valued functions on A A given by (flo,,)IJ.I, (/1 0 ,,) IJ .1 , real-valued (fto~)fJ.1 (/otp)IJ.,I; continuous, they are all sero zero (f. ° .,) IJ.I and (f 0 cp) IJ.I ; the first tint two are continuoua, outside f/J-t(D'), cp-l(D'), and they satisfy

-

«(flocp)IJ.I)(x) S (U10f/J)IJ.,I)(x) ~ «(focp)IJ.I)(x) (UoqJ)IJ.I)(x) S ~ «(f.ocp)IJ.I)(:r;) «Jtotp)IJ.I)(x) U;;;-cp)IJ.1 and (f.ocp)IJ.1 for aU xE A. Thus U;;-f/J)IJ.I Ulof/J)IJ.,I are continuous on E- and (110o ,,) IJ.. I<x) S (fo cp) IJ.I (x) S (flO cp) IJ.I (:r;) Ul ¥»IJ"I(x) ~ (jotp)IJ.I(x) (Jlotp)IJ.,I(:z:)

asaumption our theorem holds for II for all x E 1. By assumption /1 and II, /t, BO 10 that

J,( (J,o cp) IJ .1- (JIO J.I) - fA «(f,o,,)IJ.I«J, 0 cp) IJ.. 1- (flO (J10 cp) IJ.I) IJ.. I) /'«(I,ocp)IJ.I(flo cp) IIJ.I> - LCA) L(A) (fl (J, - il) ft> < •. f.

Since.E was an arbitrary positive number, the lemma implies that (10 Since U0 cp) tp) IJ.I IJ.I (focp)IJ.1 is integrable on 1. Thus U 0 tp) IJ• I is integrable on A. Furthermore, from the inequalities 0 cp) IJ.I LCA)/,,""= L(J.o (flocp)IJ .. 1S (focp)lJ .. 1S /..(A'!. cp) IJ.I ~ fA (J S fA (flocp)IJ.I(J,o cp) IJ.1- LeA)/a /"(A,!'

and

we deduce that (focp)IJ.I- LA,!I S LCA) (fl-/l) IL(Jocp)IJ.I~ /"(Al (J,-ft) < This being true for all

f

t. f.

> 0, we have I'" f~A'!' f~A,!1 fA (Jocp)IJ.. (focp)IJ.I-

proving the contention of this section of the proof. Therefore from now 8118Ume I/ to be continuoUB. continuous. on we may 888Ume

(4) If (A.I.e. fA,I.EII is a collection of open subsets aubletll of . . euch aueh Uw,t ... u.t A U A, hltriction V A. and for each ,It E 8 the theorem is true for A. and aod the l'eItricUon

aEB eE' of "f/J to

A" tp. For let/: A.. then the theorem is tN8 true for A and ". let /: ,,(A) -. - R be •a continuous continuoue function that is aero ou_de outaide the compact aublet sublet D of ,,(A). (f/J(A,) I.EII ia (,,(A.)}.e. ie a collection 0; 01 open lubleta eubeets 01 of .,(A) ,,(A) whOle whoee union ia is .-(A). .,(A). By the proposition continuoUl ~II: ... B- -+ continuoue functiolll functioDl #1) ••• , '/I,,: - (0, 1) may be found euch ",,,(Z) - 1 lor each •zED aod each '/I, ie luch that '/Il(Z) ~l(%) '/I.(z) E D and #. i. sera subset of .,(A.(t"), ,,(A'h,», for lOme .(t) E 8. For each zero outside a compact sublet i -= ... 1, .. ..._,N , N we have

"'h ... ,

+ ·.. ... +

f.(A) '/1;/-. ~;J -. £
£(A,(I)

0

0 ,,)

N

1_1 i-I

1f.

" 1-1 E~J«tl'4)otp)IJ,,1 1 - f. E "';/" - E 1 "'~J;/ - EL" 1«"'4) IJ.,I it (fo,,)IJ.,I, .. LE ~ «'/I4)o")IJ,,I«~4)o")IJ,,I- L(fo")IJ,,I, N

.,(A) ,,(A)

N

~A) 1-1 _1 .,(A)

_I ...1

N

.,(A) "CA)

_I _I

0 ,,)

A

which was to be shown. ie true for ft n -- 1. To prove thi. true we may suppoee A (5) The theorem is to be an open interval, by (4:). I be a continuous (4). Let 1 continuoue real-valued function ,,(A) that is zero outside subset of .,(A). The function on .,(A) outaide lOme compact IlUbeet 10 tp is aero 10" zero outside lOme compact IUblet subset of A, 10 we can find CIt 0, II b E A, ,,
0"

b».

( 1-1

(1-1

h(A)' heA)'

"aN)) "Uu)

1J-

f.tJ(t)/1"<' )1- {(fo,,),,'-l (/0 tt) 1/.1. J.r (J 0 #P)'" -1 (fo")lJ,,I. ~.). f'(.)

If "f/J is decreasing the computation il is

r.

A

tfl (J ,,)fJ' - {CJ / .. 1 f.,,«."n r<-) / - f (f ,,«• J/ -- J.(t). ltJ(t). 1f. /.. J.r• CI - L(fo,,)IJ,,(· (fo")IJ,,f· a

,(A) "CA)

.tI)

)/ -

0 ,,).,' -

-fll)

0 ,,)( - " , )

...ume ..um.

TbiI proves provee the theorem for" lor n - 1. Therefore from lrom DOW now on .e we may Tbie that fa n > 1. theormo is true if for lor certain i, i-I, j - I, ... ••• , a .e we have bave (6) The theore~ ",(XI, z.) - %I. apeciaI ".(ZI, .•• , %,0) ZJ. By virtue of (1), it lU8leM eufBceI to prove WI this in the epeciaI cue case where i - j - 1, that is

t"

,,(ZI, ••• %,0), ... ••• , .,. (Zl. • •• ,,(2:1, ••••, z.) - (ZI, (%1, .,.(zs, fPI(ZI, .•• ••• , z.), ",(Zi, • •,

:r..». z.».

. . .. .

z. X.

IIUIIII.... Ilft'JICIaALa IIUL'l'IPLIIIJI'I'IICI&ALe

A is •a union JjJa, for any point 01 of A is UDion of open intervals interva.1a of E-, ia the center of an open _tirely contained in A, and an open ball of radius ,r conopeD ball that is entirely tUna open interval havilll haviOl the l&Dle lIIlme center and aides 2rl 2r/ V"ft". Thus taina the opeD

yn.

by (.) (~) we may .-une Identifying EJl't UIUJIle that A ie ia itaelf itAlelf an open interval. Identifyinl with RX ]i,'--l, we have A - B X X C, where B and C are open intervall with. X B-1, intervalt in R be.• that, •R aDd and ]i,'--l B-1 relpectively. respectively. Let /: J: .,(A) .... - B be.a continuous funotion that is ..., outaide aIt compact IUbeet aubeet of f'(A) cp(A) C B X B-1. The funodon funotion iI left) outaicle B X ]i,'--l ... R ap'eeI B-l-+ B which . . . . with IJ on fJ(A). .,(A) and otherwiIe othenriae .. II RI'O HI'O II oontinUOUl. l(8): B-l-+R ]i,'--I .... R liven by ,,,,)(,) oootioUOUl. For each ss E B, \h_ the function function/ca): Ic-)VI) -1(., ilsero B-1. hence I(s, ,) iI is continuoUl continuoua and ia IeI'O outlide. outlide a compact subeet aubeet of B-1, lntecrable on B-1, 10 we have loterabl-

1: I:

l.-a

I~)f I.tA)J - I.xro-t I.xr-d/ -- I. (/.,.-t/), (/.....·1),

1....,/(.).

where /.-.1I denota denote. the function on B whoeevalue ats at s is / ..... 1(.). For any zs E B CODIider eoDIider the function .,(~: C.... CPc-l: C - B--l B-1 which ia is defined by

Sa»

CPc.)(.zs, ... z.) - (",(s, z.), ... St, ••. ... , s.» f'(e)(St, • • .,, Sa) (~(z, St, ... ••• , Sa), • • .,, ",,(s, .,.(z,~, for all (St, ••. s.) E C. CPC-) is a one-one continuoUlly continuously differentiable map lor aU (ZI, ••• , ~) fP(II) iI whole I..". .)(a)J ••••, Sa) whoee jacobian 1 'Ic-' is ia (J (I.) '-" that is for each (St, ••• s.) E C we have .(z, St, ••• , Sa). 1J f1(fJ)(J:t, 'Ic-'(St, ••• , Sa) s.) - JI.(s, z.). Since our theorem hold! holds for ,,- 1 (by (2», we can compute (b7

I.....1lc-) - /'Ic-)cC)JC-) - Ie /Q (ff.-)ofJ<-»IJ~I(/c-)oCP(a)IJ'Ic-,I- Ie IQ «/ofJ>lJ.I)(-», (Uocp)IJ.l)ca), 1.-a f.-) -

1",,)(C)ff.-) -

10 that

Therefore (fofl)IJ.I). I.cA)J- I./. (Ie (lc(Jocp)II.I).

I~)f-

(/ocp) 11.1 is a continuoUl continuous real-valued function on B X that. is Now (jofl)IJ.1 X C that ia outaide outBide a compact subset sublet of B X C, hence integrable on B X OJ Cj aIIo, alto, for each ssEB E B the function (U 0 fI) 1.1:.1 )(8) is a & continuoUi «Jocp)IJ:.I)C-) continuous real-valued function on C that is IeI'O IeIO outside a coin pact, IUbeet of C, hence integrable coinpact.aubeet int.ear&ble the laat iterated integral intearal equals (J 0 cp) 1J" I. Thus we on C. Therefore the"'t (f0f')IJ"I. indeed have

sero IeI'O

I.xc

cue. in our epecial caa (7) We now prove the theorem. For any point

G 0

E A we hava' haVl'

J.(o) ,. 0, 10 that 1.(0) ... 0,10 that. ::: (0) ,. ... 0 for at at. least leut one i-I, ... , ft. n. For given i,

the 8ubset is not zero aero iB is open and the union of these ft1& 8Ubeet of A where 8tpa/Bz. atpalaSj iB subsets is suffices to prove the theorem theoreln for each one of these iB A, so 80 by (4) it 8uffices subsets. assume that atpalas, 8tp,./aZ,i is never zero on A. By (1), subeets. Therefore lYe may 88IIWlle we may asaume that i ... == 11., 8f/J./8z. is never zero on A. Now conmayauume R, that is iB afP.las. sider Bider the map (1: 17: A -+ - Era E- defined by

The jacobian of II iB is 8"./Bz., fa never aero, HfO, 10 by the invel'lle invene function 01 17 IJfP.IIJs., which it theorem each point CI such that G E A is iB contained in an open lubeet A. of A 8uch the restriction fronl A. onto an aD open subset reamction of (1 17 to A. is iB a one-one map from O"(A.) cr 1 : 17(A.) C7(A.) --+ A. iB is also alao continuouely continuously 17(A.) of E- and such that the map rl: differentiable. Again by (4), it suffices to prove the theorem for each A A•. •. Thus we may 888ume that the map 17: 0": A --+ E- is a one-one map frorp A _ A is also continuonto the open subset 17(A) CT(A) of E- and that rl: r 1 : 17(A) CT(A) -+ 1 map T" T == fP fJ 0 0 r r l is therefore a one-one conously differentiable. The Ill&P tinuously differentiable map from 17(A) C7(A) onto fP(A), tp(A), and fP fJ -l a TO 17. tI. The maps 17 fS and T are such that if S z, = =: (Sl, (:el, •.. ••• , s.) z.) E A, then 17(S) CT(~) = (Sl, (Zl, "', ••• , Sa-l, f).(z» tpa(s» and "(ZI, T(Sl, ... (fPl(s), •.. Sa-I, ••• ,, Sa-I, %.-1, tpa(s» -= == (C/'l(z), ••• , tpa(s». By (6), the theorem holds for the map 17 fS of A, provided J. iB is nowhere zero on A, and also for the map TT of 17(A), fT(A), providedJ~ provided J r is nowhere zero on 17(A). fS(A). Therefore for any continuous func~ion f: fP(A) ,,(A) ..... E" that is zero outside a compact func~on I: - Esubset of 'P(A) fP(A) we have, provided J. and JJ~r are nowhere zero,

"".(x»

"".(x».

l.tA)l/.(A) (J(JOT) IJ,I- IAfA «Uo1')1 «UOT)lJ.1) f.
0

=

17)IJ.1 er) IJ.I

..... IA fA (J(JofP)I(J~o17)J.I· fI) I(J,o er)J.I· 0

The theorem will therefore be proved if we can show that

J. - (J. ° 17)J. at each point of A. Por those who know linear algebra thiB this equality iB is an immediate consequence of the last paragraph of the first section of Chapter inlmediate (Bince the detemunant determinant of the product of two linear transformations is IX (since the product of the determinants), deterllunants), but it is possible to give a more "eleas followa. T00 17, mentary" proof, 88 follows. Since fP fJ = .,. CT, for auy any i, j == 1, ... ,ft , n the giVeB chain rule gives .

(f/JI); =a «TOCT)i); (Ti OCT), -= == (~); ... «T017Mi -=- (T,017),

.

t E

.-1

..

«Ti)tOO")(fTi)} «T')~017)(17,,)i

i-I

_ { { (T,); ° -= 0 17 CT + «TJ~ «"'J~ °0 17)(fP.); tI)('PII)i if jf j < ft n

«TI): «"' a>: 017)(tpa)~ CT )(~)~ 0

if j .. - ft. n.

Jf4 lIM

X. MULTIPLE IIO'LTIPLII INTBGRALB INTI!IORALB

..

Thus the n" X "n square array «/Pi);) «f,'i)/) is obtained from the" the n X "fa square equare array «or,); 0ocr) (1) 88 &8 follows: if j < " then each element of the JIA j" column of the former equals the corresponding element of the J1A j" column of the latter plus ('PtI); ftlA column of the latter, (tp,,), times the corresponding element of the "" while each element of the ,," "eA column of the former equals equala (tp,,)! (",,)~ times the corresponding element of the ,," n" column of the latter. By the elementary properties of determinants we have

«1'.);

det «'Pi)i) «/Pi)j)

CT) • (tp,,)!, ('PtI)~, = det «Ti)j «Ti)/ocr). 0

that is, J_ (J"OCT)J., J" == (J.ocr)J., which is precisely what remained to be shown.

PROBLEMS

II, ... 1. Let 11, •.. , IN be disjoint open intervals in E". Show that If if J., J I, open intervals in E" such BUch that

.. •••. ,J" , J" are

I.V ... VINCJIV .. • VJ" rIV···VINeJIU···UJ" then vol (Is) vol(Ia)

+ ... + vol (1.) (I,,) S S vol (J.) + ... + vol YOI (J (I.). II).

l'

I. . computational argumel)t &J'IUIDeJ)t for Example 38 of •, 1T 2. Can you give a 1MB continuous real-valued function on a closed interval in B3. Prove that a continuou8 .- is intep'able, using only Lemma 1 of 12 • 2 and uniform continuity. integrable, n-dimensional generalisation of Problem 6, Chapter VI, with [0, 4. Do the "-dimensional (0, b)

c1011ed interval 1 of Eft E- and replaced by a closed

L·L'J(z)dz I

(z)dz by

/,J. f,1.

• 2. 5. Write down in all detail the proof of the fint first proposition of 12.

BUbeet A of 8". E". Show that if 6. Let fI be a real-valued function on a BUbBet 10 does doee then 80 interval.)

fAII L

exiate, exiat8,

III, and ILIIII< LIII· (Him: FInt IAfA III. Firat U8UIIle that A is is.a c10aed clo8ed lIIIUDl8

(a> Let I/ be a real-valued function on a cloeed interval 1 of B-. 7. (a) Ra. Bhow Show that If if I is integrable on 1 then 80 is ia/'. JI. Let I. functions on a closed (b) u,t /, , be real-valued functioDl cloeed interval! interval I of B-. E". Bhow Show that if I and , are integrable on 1I then 10 80 ia/,. il/', functiona on a 8Ilbllet BUbeet A of 8". E". Show that if (e) Let I, , be real-valued functions

I.,

and

L,

fAI fA /

exist. then exist,

IAfA I, exists.

Cd) IAt Let J I be a real-valued function on a& BUbaet (d) subset A of B- and let B C A. Show

If that if

exists and B Bbaa has volume. volume, then 1./ f.1 exiate. IAfAII exiIta exists.

PftOBLBU8 PIKIBLZMS

145 J45

that if E" have volume, then 80 do the seta sets (e) Show t.hat jf the subsets A and B of Eft A flB, r'tB, A VB vB and A-B.

8. Show that if a subset A C Eft has ACE" baa volume, then the interior of A (cf. (of. Prob. 16, Chap. III) has baa the same volume. 9. Show that a bounded 8ubset A of E" has volume if and only if the boundary (d. PI'ob. Pl'oh. 17, Chap. III) haR volume sero. of A (cf. zero.

bounded real-valued function on a closed interval 1 of E-. E". Prove 10. Let f be a hounded that f is int(\grahle integrable on 1 if and only if, for any e, a 6 > 0, 1 is the union of a ~ubintervals such that the 8um sum of the volumes of thoee those finite set of closed ~ubintervals B. subintervals 8ubint~rval~ on which f varies by at least e is less than 6. E". Prove 11. Let. f be a bounded r('.al-valued funcUon on a cloAed cloRCd interval 1 of 8-. that, h~ integrahle on 1 if and only if, for each fe > 0, the set that. ffiR Bet of pointe points of 1I at

osdllation of If (cf. Probe Prob. 5, Chap. IV) is at least t• baa volume aero. sero. which the oscillation

preceding problem to show that a bounded real-valued function I/ on 12. US!' tJs~ the precedinp; c1oSl'd interval I of Eft Bet of pointe points of 1 E" is integrable on 1 if and only if the eet a closM continuoufl is the union of a sequence of subeets at which f i~ not continuous subeet. of 1 of volume zero. .13. Show that the nonempty subset of (0,1) (0,1\ consisting of thOle numben numbel'll which 13. havE' p-xpSn1&\iOnA none of whose digits OWD clutter have decimal derimall'xpanflionR dilits is 6 ill i8 the let of its ita on clueter poinu.. pointPI. Show that thi,. thill set is of volume sero. lero. . intl'ger nn> 8" be the union of the 0llen OllCll ball, in R of centen C8ntel'll 14. For each int.e"pr > 1 let 8,. 1 )/71 and ..adii radii 1/1&2"+1_ l/n2-+ I• Prove that U V 8. ia I, an open l/n,2/n, ... , (n - 1)/,. B.

.-.-1,1.......

subRet of [0, II (Hint: If this let set had volume, the volume 8ubRet 1) without volum('. volunle. (1Iint: But t.ht' th(' union of any finite number of 8.'s 8,,'s has baa volume less would be I. J. But. than 1/2.)

c

A Eft and let f: A --+ --0 EM. E"'. Consider the condition that there exist 801M some 15. Let ACE" MER such that d{j(x), f(y» ~ Afd(x,,,) x,,, E A. J\l E R su('h d(!(x),!(y» Ald(x,J/) for all x,YEA. (II.) ill Ratisfied flBtisfied if I/ is the restriction to A of lOme some (R) Show that the ('ondition condition iR

differentiable map into EM E'" of some open subset of EE" containing A, if the partial derivatives of the component functions of I/ are bounded on A and A ('ontainR the entire line ll&egment S!'gment between any two of ita its pointe. cant-ainR t.he and vol (A) - 0, then (b) Show that if the condition is satisfied, if m - ft, aDd vol (I (f(A» >= O. (Hint: A is contained in the union of a finite finita Dumber of (A» == total volume less than any prescribed positive Dumber.) number.) cubes of tot.al (c) Show that if the condition is satisfied, if m > ft, n, and A ie is bounded then

vol (J(A» (f(A» .. - o. O. (Contrast with Prob. 31, Chap. IV.)

16. Prove that if A C Eft has positive volume and I/ is a positive-valued function ACE" functioD

fA

fA

exists, then J f > O. (Hi"t: (Hint: Reduce to the ease caae where A is on A such that fJ exi8ts, lex) ~ el) .1) - 0, a cloRed interval and for any positive e we have vol (Ix E A : fez) use compactness.) then try to usc AcE" f: A -+ --0 R a continuous funetion. function. Show 17. Let A C Era be a set with volume and J: that. if the S!'t 01J has volume zero, then the set Bet IIxx E A : that, set IIxx E A : If(x) (x) = 0 f(z) > 01 has volume.

M6 1M

MULTIPLIIINTBOBALB X. .VLTJ'~ IIf'I'BOIlAUI

18. Let A be a bolUlded bounded aublet subeet of E" and II, I., It, I., /., . .. a sequence B" aDd eequence of real-valued f. Show that if functions on A that converges uniformly to the limit function I. funetions ( i. exists for all nI, then ( / exists, and ( I - lim ( Ii.. this true if A I. tn, •. Is [h.t .... J.. h.t hJ. .•.~h is not bounded?

11

II. 1M B- be compact and aDd have 'VOlume, B be open, and let I be •a Let A C Era volume, let U C R 19. CODtinuoul real-valued fUDctioD OIl on the let ClODUnUOUS real-vaIued functioD eet f(SI, ,s.,,) eB-+l: eA"e UI. I(SI,.... ",Z.,.> EB-+l: (Sa, (ZI, ... ..• ,s.) ,Za) EA"e UJ. Prove that if '1/'" IJfI~ uiata is ClObtinuoua continuoul OIl on the th8 latter eet, let, then uiate and ill

!~ 1/(Z, il(S, ,),) a - iI~ ~

20. m.

(s, ,)

a.

!At V compact .Dd NUl K be continuoua CODtinuoua .-1Let. v C .B-. be 00IIlpaCt aDd have volume and let A aDd naIvalued fUDCtio_ fUOCUoDa OIl on V and V X V reapectively. respectively. Show that if

Ivol (V) K(s, K(z, r) I !vol

<1

e

aU s, , E V then there ill is a unique continuoul continuous real-valued function" on V lor all lunction " OIl IUch IUCh that

rp(s) .,(z) - A(s)

+ Iv K(z, ,) rp(r) tN) tlfI dr

forallse lor all ~ E V.

/(s, ,) II) - 0 if :r % aDd and 1/fI are not. Dot both F: (0, 1) X (0, 1) .... - R B be defined by I(s, rational, while !(z,,) 1/9 if s aad and , are rational and q9 is the aunallest 1(S, ,) - 1/, amaIlest posip0sisuch that tp is an aD intepr. integer. Show that tive intepr BUcb fZ ill

21. Let

1... .

a

1b<... /(z, ,) t&tlr dr 1Ix ... /(S,,)

- 0 but

/ ,) tlr) 1,,11 • ) a 1 ....(/"'1\ (f"'II /(s, (S,')

I/"'....11 (/... (/"'11 I (:r, ,) a) tlfl? tltl? 1 /~.... _a+w+~. s". .. a tlr .. 111~""_"""09s",,,a.,,

ill de&oed. What about Is not defined.

22. Compute 2t.

I\/(S, ,)

23. Cbanp Change the order of intepatiOD integraUoD in

r. (/..... (J.~.I'"(J.'lI.........1'I'J,4. /(z, r, C ,'/1(/.'-.........

/(:&, fI, ,) z)

tlz) dr) d:&a dz) dtI)

aoawera). (five aDlWen).

24. Show U (a, (CI, 6J iI intervalln R and I: (0, II. 8bcnr that if ill •a cloeed cloaed interval In It (a, 61

uou,thea uoua, tlaeD . II.. Compute

r.J: (t

I(:r, fI) /(%, ,)

d,) atis -

r.J: (L'/(Z, (I.'

fez, II) y)

II) ..... R is contlncontinX (a, (G, 61B ..

dJ:) dll. tU) d/l.

e" :

({ (Zit ••• , s.) Ell-: 0 0 < :r" Za, ••• ,:r , z".. and :r, Sa + vol «((:r"

·.· + z" S :S 11). ... +:r"

28. closed unit. unit ball in B-, E-, that is, the set t8. Let V. be the volume of the clot!ed

H:r., ... +~I t (SI, ... • •• ,,~) Sa) eB-: E E- : :r,I+ Sa' + ··· + %a' S 11. 1 J• 1 if ,. > 1 then V. ....... /.' /.' (l (1 - ')'8-1 tI)I"-'I/' Show that if" v. -- 2 Vto-! /1 dt, Ill, and bence hence (applying V".. --= (21f/R)V,,--a fa Prob. 39, Chap. VII) that V (211'/ft) V...... if ft

> 2.

PBOBLBMS PBOBLB118

247 ~7

'D. Let A C E·, BeE", B C E-, let J and ,fI be integrable real-valued functions on A aDd and c E-, B respectively, and let 1r 1r..A and 1r. 1r. be the I>rojections projections of AX B onto its factors, that is r1r..A(X,1I) ...• (x, 11) ... == 1/11 if Zx E A and 11 E B. Show that (z,1I) ,. zX and "'.(Z,II) :II:

Lx.

(j T.)) -(J 0 TA)(g 1r..}{fJ 0 ....

(£/)(/.(/)· (£/)(/. (I).

(Hint: Problem 7 can simplify the proof.)

28. Let ACE" g-. Show that ACE- and B C 8-. (a) if A and B have volume, then (by Problem 27) vol (A X B) -=- vol (A) vol (B) (b) if vol (A X B) exists exi&ta and is nOlllero, nonaero, then A and B have volume (0) (c) if vol (A X B) - 0 then A or B baa volume lero. thlt under the conditions of the change of variable theorem, ip 29. Prove that ~ maps any subset cOlnpact liUbret l:tub3et of A and has volume aubaet of A tha.t that is contained t.'Ontained in a compact onto a 8ubset subset of .,(A) ,,(A) that has volume. is a real-valued function on E' such that. 30. Prove that if J it

Ir/

(z,1I) fk dz dll d1l 1,/(:&,11)

..

1,1 Irl exilts,

exists, then

1~.t9S... /(r COl ItSr.O!>~ COl 6, 8, r sin lin 6) 8) r dr dB.

(Hint: First prove this some open subset of E' J:I containing the thil if I is zero on lOme z-axis. Problem 7(d) can help in l)assin~ positive x-axis. J>assUl~ to the general case.)

> 0 we have e-·I --, dz dx dll dy < e-al- w• dz dx d/l dy r.a-". ,-all-Wi a.velO.~J

(a) Use Ulle Problem 30 to show that for any Iek 31.
l

/.tSr9 ~II 8S'Srll

e-r' r dr dB ... =1 .{ ,-rI }o<.., ~-'~ .1""-'91

1L

··wElO.tl

< ( }o<...

,-ao-we-zl- w3 dJ: dx dy -

}o
aT+wlSMI .T+Wll94"

,-rI ,

_ e-t4 , dr {( d, dB. oS-S_1I oS,S·1I

}D
(b) Deduce that

1(1rtl) «J." < (Ie'r th}1 < i( rW). ~(l - ,-to) r ddx)1 ~(11 -- rill)' d

e-··

(+v'r '\IT (c) Prove that J. e-al dx that}. dz - 2 (cf. Prob. 28, Chap. VI). 32. Let A be an open subset of E- and 1 J a real-valued fuuction function on A. Let :D 5) be the llet of 01 compact tlub8etas lIubaebs of A th"t th&t have volume. let volulne. Cadi Cedi J I abaolutel/l abaolutelll integrable on

ID exiatl for each D E :D fD 1exists 5) and there is a number nwnber L E R such lueh that for any exilts some lOme D E :D web that if D' E :D and D' ::::> • > 0 there exists 5) such nand :J D then

A if

IID,f - I< L

4!.

(a) Show that if L exists, it is unique. (Hence we may write (a>

L ... (b) Show that if

that that.

IA 1 L

exists, then

1;""1.) I;"" I.)

. /. I: 1= 1= 1 /..1.

ID 1I exists for all D E :D.) 5).)

(IIi,,,: (lIi,d: Problem 7(d) 7(11) impliCij iDlJlliell

148 M

X. IIULTIPLB IIUIIl'I.LII INTIJOBALI UIftOBAJA

/D

that if /DI1 existe D E 5), then 1 abeolutely intepoable integrable on exists for all DEI), I illHI ablolutel)' A if and only if for any • > 0 there exiate exiIte IOII1fI IOID8 DEI) DES) IIleh lUcia tIW that if 1)' e I)S) D' E

(0) Show

and D' cA C A - D then

1.1 :

I /D,/I< -' /D,J I< e, which is true if and only if the lit

I/D 1: DDEI) e 5)1isHI bounded, and this in tum IIis true if aDd ADd onl7 only if If the lit -' I/f /.D111III : De 5)1 is bounded. (Note that JD III .... exiatI 'for aD Del), De 5), bJ Del)I 01' all Problem 6.) 8.>

I.

I;

(d) Show that if 1 exiIt8. f ill HI continUOUll continUOUl and IJ and A are bounded then / ; I am.. (e) 6) be an open interval ill be. continuoUl rea1-valued real-valued <e> Let (a, (a,6) in R and let IJ be a continuoue function on I z E R : aG < z% S 6 J• Show that if }c...) fc'" f/ alate exiata then S 6J.

l: 1_ J.' fez) (}e_ 1-1' I(~) dz (01. (cl. Prob. Probe 71, Chap. VI), and that iff ill takes tak. on only nonnegative th uiIta then t: existe. nonneptive values and / ; f(z) dz J:continuous f.u.iIta. Let t E R: z > aJ,J and let fI be a contiDuoua Letaa e E R, A - Iz real-valued ; J exists existe then / ; ffunction runction on I z E R : z ~ al. Show that if / /;1 J.. . thdz (d.(01. Prob. 211,28, Chap. VI), and that IfillI takes tak. on only oaly __ . neptive th exists ; J exiate. exitIta. nepUve values aDd and /..... /..... fez) dz exiate then / /;1 . {.... sin L . . sin ~) 8how Show that J.+-: z . dz exiata, Jo z exmts, but / : izi: z cis liz does not. (Yo

}(

dz

a+ e+

)

OD

I(z)

(f) (I)

ft,

1m

R :

%

%

Q

1

exi8tI

J

R :

/..... I(z) /(Z)

DOD-

I(z)

(I)

%

•

%

of

Suggestions for Further Reading

McGraw-Hill Book Ahlfors, Lars V. Compln Compk~ Analym. New York: McGraw-HUI FAfition, 1966. Company, 1953; Second Edit·ion, Apostol, to ~ Apllstol, 'rOlll Tom M. MaUaemtJl,ical Mathematicol JinGlylli,: Analyai.: a modm-n modern apprOtlCla approacl& Co calculu•. Reading, Mus. Ms..:: Addison-Wesley Publishing Company, Inc., calculus. 1957. Bart.le, Robert G. The Element. oj Real Analysi,. Analyai•. New York: John Wiley & Sons, Inc., 1964. oj Real Fu"ctiOftl. Fuftdione. New York: John Wiley. Wiley 4: Boas, Ralph P., Jr. A Primer 0/ Bonl, Bons, Inc., 1960. Buck, R. Creighton. Advanced Calculus. CalculUi. New York: MeOraw-Hill MeGraw-Hill Book 1965.. .. Company, 1956; Second Edition, 1965 Dicudonn~, Anal,,_•. New York: Academic Dicudonn~, Jean. Foundationa Foundotiona 01 0/ Modem Analysia. PreBS, 1960. Pre88, Gleason, Andrew M. Fundamenlala FundGmenlGl, 0/ oj Ab.,CJCt AbetTact AftClZ1IftI. Anal"... JteadiDa, Readinc. M_.: M ... : Addison-Wesley Publishing Company, Inc., 1966. oj Real A Anal"si•. New York: Blaildell BIaiedeU Goldberc, Goldberl, Richard R. MeUaod. MeOwd. 01 Publishing Company, 1964. Paul R. Noiv, No'We 8" Set TMOfl/. Theory. Princeton, N.J.: D. Van Noetrand NOIltrand Halmos, P"ul Company, Inc., 1960. Landau, Edmund. Foundationl A ntJlyai,. Foundationa 01 oj Anal"ai •. New York: Chellea Publiahinl Publiahing Company, 1951. Royden, H. L. Real Analyaia. A1UJl,lft,. New York: The Macmillan Company, 1963. oj Mat1&emtJtictJl MatMmatical AMI,••. Analyai•. New York: McGrawMcOrawRudin, Walter. Wl\l~r. Principle. Principlt. 0/ Hill Book Company, 1953; Second Edition, 1964. 19M. Co ",.,..., Spivak, Michae.1. CalculUi OR MGfI,i/oltP: Mani./oltU: • modem appreaeA approcreA 10 dtutict;al tlaeorem. 01 oj adwJnced c:alculUi. New York: W. A. Benjamin, Benjamm, IDe 1985. tIa,orem. tJt.lNnced calculua. IDe•••• 1985"

_"ai,.

Index

Ablolute AbIOIute value, 22, 13 03 Abeoiuteiy con..... _verpatt ... ..... 144 Abeolutely i_, 1" Abeolutely AbIOIutely iDtep"&bIe, iDteanbIe, 247 ~7 Addition, AcIctitioD, 18 Additiye Additive my__, iDv.., 18 18 Agrepte,2 Agrept.e,2 Algebra, AIpbra, fundamental theorem of, 165 1M Altematmll8fiee, AlterDatia,..-. 145 Antlderivative,l27 ADtiderivative, 127 Arcwiae Arcwlae connected, 93 Area, 112, 113, m 224 Auociativlty, Auooiativlty, 18 Ball,87 Ball, 87 ebed, 87 c1oeed, opeD, 37 87 Between, 80 Binomial ...,183 I8rieI, 183 theorem, 110 Bound . .ted .ten Jow., lower, 21

_

1eut Ieut upper, 18 D 10. .,25 ,21 10. upper, 23 D

Boundary, 82

Bounded fWlCtioD, 78 fUlled_, lIquenoe, .7 ~'7 IIt,Q -t,G totally, 81 t4taUy, 86

Calculue, fundamental luadamental dJeorem Calculus, theorem of, 128 Cart.-iua product, produot, 7 Cart.ian Cauchy mean _ _ value value theorem, theorem, 109 Cauchy 109 1Iq1IIIlOII, II Cauchy 18qU8DC8, al 87 Center, 31 CltaiD rule, 102-103, 102-101, 188, 1t18, .1 Chain 201 ChaDp of 01 variable in intqration, intqrat.ioD, 128, 239 Cbanp ClO88Cl C10eed baII,87 balI,37 iDterva1, 38, 88, 43 t3 iDWYal, .t, «) 40 let, CIoIure, U C1oIure,82 a..- poiat, II QUIter U CaIleatiaD, 2 CoIIectioD, Commutativity, 18

Compact Com)llld apace, metric ip&OI,

a.

~ l8quentially, 81 86 I8qU8lltially,

Bet,M lit, It Compariloo t.-t, teet, Compuiloo Complement, 65 Complete metric ..,.., .pace, 62 Complet.e Complex Dormed vector 1J*I8, apaoe, ~ 83~ DOI'med

1"

DUmben, 30 Dumben,30

number a)'8tem, Iyatem, 30 Dumber ComponeDt function, 77 Component function, 9 Compoeed funoticm, ComPOJition 01 of fUDClt.ioDa, functiou, 9 eo...po.ition Connected metric ..,.., 69 apace, 19 eet,69 Coutant Ccmatant functioD, 8tI 89 ContinUOUl function, IUDCtIaa, 88 uailarmly, uniformly, 80 ContinUClUlly . . .t4bIe, 198 . Continuoualy diI dilerent4bte, Contraction map, 171

....

COnverl8,4a,83,141 CODY8rII,41,88,I'l

absolutely, 144, 160 1&0 at a& point, 141 86, 1'1 14:1 uniformly, 81, Convergence CoDY8rIIDC8 iDterval Interval 01, In 112 radiUl racliUl 01, of, In 162 CoDverpDt CoDY8rIIDt lIquenoe,46 I8qU8llCll, 41 I8qU8llCll lequence 01 of funot.lolUl. funotlou. 88 --.141 . . . . 141 CoDvex Conyex funotion, function, 110 CorreIpoad, Correapond, 8 CorreIpcmde_,oM-Oll8,10 Correepondenoe, one-one, 10 CoIeaant, CoIecant, 103 183 CoIiDll, Coeine, 117 157 Cotanpnt, 103 163 Curve, 83, 1M 134 1eqth0l 1eDPh 01 .. a, 1~ ..,..,.1iIliJII, apace-fillin&, 1M 94

DeoImal Deoimal expuwiOD,27 expuaion, '11 8DitAt,28 fiDitAt,28 iD8aite,27 iDfiaite,27 periodic, 31

_ lSI

IlfDD INDU

Decreuinl function, funetion, 105 eequence, 50 aaqllellClt, 110

Deue,93 Deue, 93

Derivative, 98, 100 partial, 169, IS9, UM eecond, I8CIOIId, tblrd, third, 106 108 IeCODd, IIOOnd, third order partial, 201 Diameter, 92 Differentiable, DItr_tlable, 98, 193, 196, 212 at a point, 98, 196 continuously, 198 fttimee,l06 "ti-,l08 OD OIl •a let, lilt, 100 DifFerential Dltrerential equation, 17711. 177ff. Differential operator, 203 Differentiation, Dltrerentiation, 98 under the &he intep'alsilll, int.eKralllip, 159, 169, 2f6 UDder 246

Diejoint leta, 6 Dlejoint",8 DiItance, DIetanee, 3. 34 between functioD8, 95 functiona, 87-90, 91i from aa point to aa let, 92 Diatributimy, Diltributivity, 18 Diftrp, Diverp, 141 1~1 Diverpnt eeriee, Dinrpnt - . 1.1 1'1 Dummy ftriable, ftI'Iable, 48, 113

Function (cont.) lOS increMinl, increui .... 106 integrable, 112, 218 216 Intepable, invel'le, 10, 178, 176, 211 Invene,IO, limit, 83 Omit 01, of, 72, 73 limit linear, 99 loprithmlc,l28 loprlthmio, 128 differentiable, 111&-187 UII-IIT nowhere ditrerentiable, one-Gne,one-to-oDe, one-one, ~to-oDe, 10 onto, 10 ltep, 70, 119, 222 Rep, decreaainl, 1011 106 Itrietly dlcreuinlo Itrietly IncreuInIo inereuinlt 1011 106 I&rictIy ~lOnolDetnc, 1&7, 164 tripnometrio, lli7, 183, 1M continuous. 80 uniformly continUO\ll, Fundamental theorem of alaebra, algebra, 185 01 of calculUl, calculus, 128 126 01

Ii_,.

eeries, 142 Geometric lllriel, 1<&2 8,9 Graph, 8, 9 Greater than, 19 GreateIt lower bound, 2Ii Greatest 2li

Harmonic eeriea, - . 143 Ita Element, 2 neutral, 18 Empty lilt, •, Emptyeet, EnMmble,2 Eneemble,2 EucHdean apace, Euelidean ~, 3. eonatant, 134 Euler coutant, Exponent., Exponent, exponential, 21-22 Exponential function, 129 complex,IM 1M complex, 01 an interval, 38, 218-217 Extremitiee of

3'

Family,2 Field, 18

Finite 28 decimal, 26

18"

18\ 11 11 Fixed point, 171

tbeorem, 170 theorem, Funetion,8 Function, 8 bounded, 78 eomponeat, 77 eompoDfJDt, eompolllld, 9 compoeed, COIIBtaat,89 eonetaat, 89 contin-. 88 continuous, 68 tIOIltinuoully differentiable, 198 ecmtinuOUlly COII-mc, 110 eoDveX, 1011 deereuin& 105 differentiable, 98, 100, 195 ezponeot.ial,l28 ex:ponential, 129 ideatity, 10, 10,70 identity, 70 implicit, implicit, 174, 17" -206

Identity function, 10,70 10, 70 Imap, 10 lIMP, Implicit function funetion theorem, 17',_ 174, 205 Improper int.eKral, inte....l. 131, 136, 181i, 161, 2f8 248

Increulnl InonuInl

functioa, function, 105 106 eequence, ao lllClueDOe,1IO Indlxinl: family, 8e Indexinl

Indlcel, 8 Inme-.8

InllDWD, ID&mum, 2Ii 25 InlDite Infinite decimaI,27 decimal,27

aequence, eequence, 11 ....,141 eerieI, 141 lilt, let, 11 Intepr,21 Int8pr,21 )IOIitive, poeitive, 10 Intearable, 112, 218, Intepoable, 218. 2M 224

abeoIutely, abIIoIutely, ~7 247 LeIJeIpe, Lebelpe, 118 IUlmnann,112,218 Riemann, 112,218 Inteanl, Intearal, 112, 218

improper, 135, 136, 185,2f8 186,248

iterated, 231 LebeIpe, Lebelpe, 118 116 Riemaun, Riemann, 112, 218 Intepal equation, 178, 181, 192 Intqral InteKrai toteit'd teIt, teet, 181 161 Intaior,82 Int.ior,82 point, 82. 62,

.MD. INDD

Intermediate value theorem, 82, 83 Int.erM!etion, 4, 6 In~tion,4,6 Interval c1oeed, 38, 43 clc.d, half-open, 42 ha1f-open, of converpnoe, IS2 U52 open, 38, a 43 open,88, In. . . In.... additive, 11 16 function, 10, 176, 178, 211 Imap, 10 bnap, muitipHcative, 16 multiplicative, intepoal, 231 Iterated inteBral,

Jacobian, 211 determinant, 211 determiDant, matrix, 200 Jordan meuure, 224

KroDecker delta, 208

Leut upper bound, 23 Lebelpe Lebee&ue

inf.ep'able, 116 intqrable, intesraJ,116 iDtesral, 116 LeDIth, Len&th, 224 of •a curve, 1M Lela than, 10 IA.thu, L'Hoepital'. rule, 109-110 Umit Limit functlon,83 fUDation, 83 Inferior, inferior, 83 63 of • function, 72-73 of a-eequence, • -lCquence, 45 46 IUperior,83 IUperior, 83 Line eepleDt IlellMDt between bet_ two pointe, pointl, 83 Linear difFerential equation, 189, 190 100 functJon, 99 09 function, tranlformation, 93 tnIUIformatioD, LiplChiti oondition, COIldition, 178, 183 Lipeohita Lopritbmle function, 128 Loprithmic Lo_ 25 Lower bound, 26

Metric apace, M compact" It eompIete, 62

00DDeCted, •

miD,_

Minimum, 78 MODOtoDio IIqUlllOl, 10 MultlpUeatloD, 11 MultipUeatlw la~ II Natural DUID_, 21 number, 10. 11 ...........&1 apace, M tHIimeDIional EucIicIeu EucIideua ...... HimeDIionaI HiIll8llliona1 volume, 1M

Neptlve, III Neptive,19 N_ted lilt propen",. properi7, II N.tecleet Neutral elementAl, elementl, 1. NewtoD'. Newt.oe'. method, metbod, 1_ Norm, 63 N011Il,83 Normed vector ...... apace, 63 II Nowhere diftereratlable dilFerentiable 1\1DCItIae, lee- lI7•. l67. f\1llGtlae, 118tt-tuple, 11 a-tuple, II

1.1.

Number, 1. II eompIu,ao aompIu, 80 natural, 10. 10, 21 DMural, neptift,11 neptive, 10 oI e1emeDta _ _tI la ia •a -. . 11 11 of pcIIIitive, II 10 poelti"' ratloDal, 21 ratloul, real, 18 16 Objeat, 2 I Objea\ One, II On"16

<>-one, ... to-oDe, 10 One-oae. .to-oae, Oato,10 Onto, 10 Opea ball, 37 iaterval, 18,43 iD....... ·a

.... -.89

Order, 10

Ordered 0rcWed pair, 7 0eeIDati0a, 11 01 0IciIIati0a, Partial

deriva"", 111, 1M IUID,

Map, Map,S8

contraction, 171 COIltraetion, MappiD& Mappin" 8 max, 24 mu,M

Maximum, 78 Mean .a1ue nlue theorem, 105 Cauchy, 109 intepals, 133 for intepU, IIIITeral ftl'iablel, 2CK-205 variablel, 204-205 for ••eral Member,2 Member, 2 Metric, M

. . III

Putitioa

141

. . . . . iD..... 111, III • UDity, 235 PeriocIIo cIeoImaI. 81 ~M

ol__,11 ked, 171 .......,12 Polar ooordinat8, 2l1,K7 PGlitift,

1.

."-,10

Bet, (ccat.) Set. (eoat.) CICIDIII!CtecI, 69 CODDeCted, diajoinf" diajoil¢, 6

RadiuI, 87 fR RadiUI, fI . . ........ . . . . . IU ~eoa l52 RatIo_1M ............. lWIaDal .......... 11

..."'..,141 ..... .,..,11 .......,..,1. .auaber,lI ..... II

.......

BIItiriatioa, I• BIItrietIaa, R.ieIMDD iDtesrabJe. 218 iD........ 112, l11,211 ...... iD...... 112,11. U2,21. . . , 112.111 -.112,21' Rolle', tIIeonm, 1M Rolle'. tbeanm, leN Roo&tM,l82 RocK teat, 112

empty, ,4 empty,. iIlit.e,ll lake, 11 ia&Dite,11 iDflDite, 11 opeo,:31 opea,.39 SiDe, 167 8pace-fiIUDI aurve. curve, M IJpaee-&1IIDa Square fOO\ 28 118 8qwn Step fllllClb, 70.119, TO, UI, 122 222 ~fuaotloD, INrIia(I formula, 181 8~form"181 IVlodt 8VlatIt cIeareuiDa. 101 106 Strictq IDenuiDa. 105 106 BtrictIJ Subialernl, SubiDt4'Ya1, 11 21..217 Sut.queaae. 46 Su"""", 48 •Sublet, .... a proper,4 proper... 8ub8pace, 38 8ubBpace, 31 Suoc.Ilve s.-ive approximation, approllimaUoo, 109, 119, 170 ~ a 1MIriee, Sum 01 eeriee, 1.1 141 Supremum, 25 Supremum.

roo"

Tr.qeat, 99, 91, 118 108 Tupat, Taylor 1eI'iea, eeriee, 166 1M 21M Taylor', theorem, 107, 2(K Taylor'. bouDjled, 65 66 Totally bouDsted, TranaitiYity(20 TrauitAYityf20 'I'rir.qle iDequality, M M TriaDI1e 19 Trichotomy, Trichotomy,19 TripDometric Tri&oDometric fuactiODl, fuactiou, 157, 183, 108, 1M Uallorm UDlorm eoa&iDulty, 80 ' OOIltiDult,.,80

. . . . . . . . .81, .,141 eaawrpaoe, 141

ODDftI'PIlG8 iD....... 111 lIS . . . . . .08 of 01 improper ilDpfOpeI" iD...... Uallormly eoatiDuoUilunotioa, COIltiDUOUl fuocUoa, 80 UDlfonal,

UDloa,4, I6 UDioD.4, bouncl, 23 Upper bound, V r.lue 01 of a function, 8 Vatue Volume, 224 ta-dimeoaiooal, 224 ft.i It-dimeIllJioor.l, W r.IIiI' product, 166 181 Wallie' 112, 218 Width, 112,218

Zero, 18 Zero,18