Elements of Mathematics Functions of a Real Variable: Elementary Theory

343 Pages·2004·2.259 MB·English

by  Nicolas Bourbaki, Philip Spain (auth.)

ELEMENTSOFMATHEMATICS
3
Berlin
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NICOLAS BOURBAKI
ELEMENTS OF MATHEMATICS
Functions of a Real Variable
Elementary Theory
1 3
Originallypublishedas
Fonctionsd’unevariableréelle
(cid:2)cHermann,Paris,1976andNicolasBourbaki,1982
Translator
PhilipSpain
UniversityofGlasgow
DepartmentofMathematics
UniversityGardens
GlasgowG128QW
Scotland
e-mail:[e-mail]
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DOI: 10.1007/978-3-642-59315-4
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To the reader
1.TheElementsofMathematicsSeriestakesupmathematicsatthebeginning,and
givescompleteproofs.Inprinciple,itrequiresnoparticularknowledgeofmathemat-
icsonthereaders’part,butonlyacertainfamiliaritywithmathematicalreasoning
andacertaincapacityforabstractthought.Neverthelessitisdirectedespeciallyto
thosewhohaveagoodknowledgeofatleastthecontentofthefirstyearortwoofa
universitymathematicscourse.
2.Themethodofexpositionwehavechosenisaxiomatic,andnormallyproceeds
fromthegeneraltotheparticular.Thedemandsofproofimposearigorouslyfixed
orderonthesubjectmatter.Itfollowsthattheutilityofcertainconsiderationsmay
notbeimmediatelyapparenttothereaderuntillaterchaptersunlesshehasalready
afairlyextendedknowledgeofmathematics.
3. The series is divided into Books and each Book into chapters. The Books
already published, either in whole or in part, in the French edition, are listed be-
low. When an English translation is available, the corresponding English title is
mentioned between parentheses. Throughout the volume a reference indicates the
Englishedition,whenavailable,andtheFrencheditionotherwise.
The´orie des Ensembles (Theory of Sets) designated by E (Set Theory)
Alge`bre (Algebra) 1 — A (Alg)
Topologie Ge´ne´rale (General Topology) — TG (Gen. Top.)
Fonctions d’une Variable Re´elle
(Functions of a Real Variable) 2 — FVR (FRV)
Espaces Vectoriels Topologiques
(Topological Vector Spaces) — EVT (Top. Vect. Sp.)
Inte´gration — INT
Alge`bre Commutative (Commutative Algebra) 3 — AC (Comm. Alg.)
Varie´tes Diffe´rentielles et Analytiques — VAR
Groupes et Alge`bres de Lie
(Lie Groups and Lie Algebras) 4 — LIE (LIE)
The´ories Spectrales TS
In the first six Books (according to the above order), every statement in the
textassumesasknownonlythoseresultswhichhavealreadydiscussedinthesame
1 Sofar,chaptersItoVIIonlyhavebeentranslated.
2 Thisvolume!
3 Sofar,chaptersItoVIIonlyhavebeentranslated.
4 Sofar,chaptersItoIIIonlyhavebeentranslated.
VI TO THE READER
chapter,orinthepreviouschaptersorderedasfollows:E;A,chaptersItoIII;TG,
chaptersItoIII;A,fromchapterIVon;TG,fromchapterIVon;FVR;EVT;INT.
FromtheseventhBookon,thereaderwillusuallyfindapreciseindicationofits
logicalrelationshiptotheotherBooks(thefirstsixBooksbeingalwaysassumedto
beknown).
4.However,wehavesometimesinsertedexamplesinthetextwhichrefertofacts
which the reader may already know but which have not yet been discussed in the
Series.Suchexamplesareplacedbetweentwoasterisks:∗...∗.Mostreaderswill
undoubtedlyfindthattheseexampleswillhelpthemtounderstandthetext.Inother
cases,thepassagesbetween∗...∗ refertoresultswhicharediscussedelsewherein
theSeries.Wehopethereaderwillbeabletoverifytheabsenceofanyviciouscircle.
5.Thelogicalframeworkofeachchapterconsistsofthedefinitions,theaxioms,
and the theorems of the chapter. These are the parts that have mainly to be borne
in mind for subsequent use. Less important results and those which can easily be
deducedfromthetheoremsarelabelledas“propositions”,“lemmas”,“corollaries”,
“remarks”,etc.Thosewhichmaybeomittedatafirstreadingareprintedinsmall
type.Acommentaryonaparticularlyimportanttheoremappearsoccasionallyunder
thenameof“scholium”.
Toavoidtediousrepetitionsitissometimesconvenienttointroducenotationor
abbreviations which are in force only within a certain chapter or a certain section
ofachapter(forexample,inachapterwhichisconcernedonlywithcommutative
rings,theword“ring”wouldalwayssignify“commutativering”).Suchconventions
arealwaysexplicitlymentioned,generallyatthebeginningofthechapterinwhich
theyoccur.
6. Some passages are designed to forewarn the reader against serious errors.
Thesepassagesaresignpostedinthemarginwiththesign (“dangerousbend”).
7.TheExercisesaredesignedbothtoenablethereadertosatisfyhimselfthathe
hasdigestedthetextandtobringtohisnoticeresultswhichhavenoplaceinthetext
butwhicharenonethelessofinterest.Themostdifficultexercisesbearthesign¶.
8. In general we have adhered to the commonly accepted terminology, except
wherethereappearedtobegoodreasonsfordeviatingfromit.
9. We have made a particular effort alwaysto use rigorously correct language,
withoutsacrificingsimplicity.Asfaraspossiblewehavedrawnattentioninthetext
toabusesoflanguage,withoutwhichanymathematicaltextrunstheriskofpedantry,
nottosayunreadability.
10. Since in principle the text consists of a dogmatic exposition of a theory,
it contains in general no references to the literature. Bibliographical are gathered
together in Historical Notes. The bibliography which follows each historical note
containsingeneralonlythosebooksandoriginalmemoirswhichhavebeenofthe
greatestimportanceintheevolutionofthetheoryunderdiscussion.Itmakesnosort
ofpretencetocompleteness.
Astotheexercises,wehavenotthoughtitworthwhileingeneraltoindicatetheir
origins, since they have been taken from many different sources (original papers,
textbooks,collectionsofexercises).
TO THE READER VII
11.InthepresentBook,referencestotheorems,axioms,definitions,... aregiven
byquotingsuccessively:
–theBook(usingtheabbreviationlistedinSection3),chapterandpage,where
theycanbefound;
–thechapterandpageonlywhenreferringtothepresentBook.
TheSummariesofResultsarequotedbytotheletterR;thusSetTheory,Rsignifies
“SummaryofResultsoftheTheoryofSets”.
CONTENTS
TOTHEREADER ............................................ V
CONTENTS ................................................. IX
CHAPTERI DERIVATIVES
§1. FIRSTDERIVATIVE ...................................... 3
1. Derivativeofavectorfunction ............................. 3
2. Linearityofdifferentiation ................................ 5
3. Derivativeofaproduct ................................... 6
4. Derivativeoftheinverseofafunction ....................... 8
5. Derivativeofacompositefunction ......................... 9
6. Derivativeofaninversefunction ........................... 9
7. Derivativesofreal-valuedfunctions ........................ 10
§2. THEMEANVALUETHEOREM ............................ 12
1. Rolle’sTheorem ........................................ 12
2. Themeanvaluetheoremforreal-valuedfunctions ............. 13
3. Themeanvaluetheoremforvectorfunctions ................. 15
4. Continuityofderivatives ................................. 18
§3. DERIVATIVESOFHIGHERORDER ........................ 20
1. Derivativesofordern ................................... 20
2. Taylor’sformula ........................................ 21
§4. CONVEXFUNCTIONSOFAREALVARIABLE .............. 23
1. Definitionofaconvexfunction ............................ 24
2. Familiesofconvexfunctions .............................. 27
3. Continuityanddifferentiabilityofconvexfunctions ............ 27
4. Criteriaforconvexity .................................... 30
Exerciseson§1 ................................................ 35
Exerciseson§2 ................................................ 37
X CONTENTS
Exerciseson§3 ................................................ 39
Exerciseson§4 ................................................ 45
CHAPTERII PRIMITIVESANDINTEGRALS
§1. PRIMITIVESANDINTEGRALS ............................ 51
1. Definitionofprimitives .................................. 51
2. Existenceofprimitives ................................... 52
3. Regulatedfunctions ..................................... 53
4. Integrals .............................................. 56
5. Propertiesofintegrals ................................... 59
6. IntegralformulafortheremainderinTaylor’sformula;
primitivesofhigherorder ................................ 62
§2. INTEGRALSOVERNON-COMPACTINTERVALS ........... 62
1. Definitionofanintegraloveranon-compactinterval ........... 62
2. Integralsofpositivefunctionsoveranon-compactinterval ...... 66
3. Absolutelyconvergentintegrals ............................ 67
§3. DERIVATIVESANDINTEGRALS
OFFUNCTIONSDEPENDINGONAPARAMETER ........... 68
1. Integralofalimitoffunctionsonacompactinterval ........... 68
2. Integralofalimitoffunctionsonanon-compactinterval ....... 69
3. Normallyconvergentintegrals ............................. 72
4. Derivativewithrespecttoaparameterofanintegral
overacompactinterval .................................. 73
5. Derivativewithrespecttoaparameterofanintegral
overanon-compactinterval ............................... 75
6. Changeoforderofintegration ............................. 76
Exerciseson§1 ................................................ 79
Exerciseson§2 ................................................ 84
Exerciseson§3 ................................................ 86
CHAPTERIII ELEMENTARYFUNCTIONS
§1. DERIVATIVESOFTHEEXPONENTIAL
ANDCIRCULARFUNCTIONS ............................. 91
1. Derivativesoftheexponentialfunctions;thenumbere .......... 91
2. Derivativeoflog x ..................................... 93
a
3. Derivativesofthecircularfunctions;thenumberπ ............ 94
4. Inversecircularfunctions ................................. 95