Welcome to Real Analysis: Continuity and Calculus, Distance and Dynamics

375 Pages·2022·29.309 MB·English

by  Benjamin B. Kennedy

AMS / MAA TEXTBOOKS VOL 70
Welcome to Real Analysis
VOL
Continuity and Calculus, Distance and Dynamics
AMS / MAA TEXTBOOKS 70
Benjamin B. Kennedy
Welcome to Real Analysis is designed for use in an introductory
undergraduate course in real analysis. Much of the development is in
the setting of the general metric space. The book makes substantial
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use not only of the real line and n -dimensional Euclidean space, but nn
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also sequence and function spaces. Proving and extending results aul
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from single-variable calculus provides motivation throughout. The esy of
more abstract ideas come to life in meaningful and accessible appli- ourt W
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cations. For example, the contraction mapping principle is used to ot e
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prove an existence and uniqueness theorem for solutions of ordinary c
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differential equations and the existence of certain fractals; the continuity of the integration
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operator on the space of continuous functions on a compact interval paves the way for some e
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results about power series. ja e
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The exposition is exceedingly clear and well-motivated. There are a wide variety of exercises i o
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and many pedagogical innovations. For example, each chapter includes Reading Questions B R
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so that students can check their understanding. In addition to the standard material in a fi rst K e
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real analysis course, the book contains two concluding chapters on dynamical systems and n
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fractals as an illustration of the power of the theory developed. ne A
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For additional information
and updates on this book, visit
www.ams.org/bookpages/text-70
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4-Color Process 374 pages on 50lb stock • Softcover • Spine 3/4”

Welcome to Real Analysis:
Continuity and Calculus,
Distance and Dynamics

AMS/MAA TEXTBOOKS
VOL 70
Welcome to Real Analysis:
Continuity and Calculus,
Distance and Dynamics
Benjamin B. Kennedy
MAATextbooksEditorialBoard
WilliamR.Green,Co-Editor
SuzanneLynneLarson,Co-Editor
PaulT.Allen MarkBollman SusanCrook
HughN.Howards WilliamJohnston EmekKose
MichaelJ.McAsey ThomasC.Ratliff PamelaRichardson
RonTaylor RuthVanderpool ErikaWard
ElizabethWilcox
2020MathematicsSubjectClassification.Primary26-01,37-01,39-01.
Foradditionalinformationandupdatesonthisbook,visit
www.ams.org/bookpages/text-70
LibraryofCongressCataloging-in-PublicationData
Names:Kennedy,BenjaminB.,author.
Title:Welcometorealanalysis:continuityandcalculus,distanceanddynamics/BenjaminB.Kennedy.
Description: Providence,RhodeIsland: MAAPress,animprintoftheAmericanMathematicalSociety,
[2021]|Series: AMS/MAAtextbooks,2577-1205;Volume70|Includesbibliographicalreferencesand
index.
Identifiers:LCCN2021036922|ISBN9781470464547(paperback)|978-1-4704-6847-7(ebook)
Subjects:LCSH:Mathematicalanalysis.|Functionsofrealvariables.|AMS:Realfunctions–Instructional
exposition(textbooks,tutorialpapers,etc.).|Dynamicalsystemsandergodictheory–Instructionalexpo-
sition(textbooks,tutorialpapers,etc.).|Differenceandfunctionalequations–Instructionalexposition
(textbooks,tutorialpapers,etc.).
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Contents
Preface xi
0 WhereWe’reStartingandWhereWe’reGoing 1
1 EssentialTools 5
1.1 Setsandstatements 5
1.2 Functions 9
1.3 Countabilityanduncountability 18
1.4 Induction 23
1.5 Orderintherealline 26
1.6 Somevitalinequalities 32
1.7 Exercises 36
2 MetricSpaces 41
2.1 Thedefinitionofametricspace 41
2.2 Importantmetricsinℝ𝑛 45
2.3 Openballsandopensetsinmetricspaces 48
2.4 Closedsetsandlimitpoints 53
2.5 Interior,closure,andboundary 59
2.6 Densesubsets 62
2.7 Equivalentmetrics 63
2.8 Normedvectorspaces 68
2.9 Abriefnoteaboutconventions 71
2.10 Exercises 71
3 Sequences 77
3.1 Convergence 78
3.2 Discretedynamicalsystems 87
3.3 Sequencesandlimitpoints 92
3.4 Algebraictheoremsforsequences 95
3.5 Subsequences 101
3.6 Completeness 107
3.7 Thecontractionmappingprinciple 110
3.8 Setsofsequencesasmetricspaces 119
3.9 Exercises 125
4 Continuity 131
4.1 Thedefinitionofcontinuity 131
vii
viii Contents
4.2 Equivalentformulationsofcontinuity 142
4.3 Continuity and limit theorems for scalar-
valuedfunctions 146
4.4 Continuityandproductsofmetricspaces 150
4.5 Uniformcontinuity 154
4.6 Themetricspace𝐶([𝑎,𝑏],ℝ) 160
4.7 Anapplicationtofunctionalequations 167
4.8 Exercises 170
5 CompactnessandConnectedness 177
5.1 Basicdefinitionsandresultsoncompactness 177
5.2 Thenestedsetpropertyforcompactsets 181
5.3 Compactnessandcontinuity 183
5.4 Otherfactsaboutcompactness 185
5.5 Connectedness 191
5.6 Periodicpointsofmapsonintervals 195
5.7 Injective continuous functions defined on
intervals 200
5.8 Exercises 202
6 TheDerivative 209
6.1 Thedefinitionofthederivative 209
6.2 Differentiationrules 218
6.3 Applicationsofthederivative 222
6.4 Exercises 229
7 TheRiemannIntegral 235
7.1 Partitionsandthedefinitionoftheintegral 235
7.2 Basicpropertiesoftheintegral 243
7.3 Thefundamentaltheoremofcalculus 247
7.4 Ordinarydifferentialequations 253
7.5 Exercises 257
8 SequencesofFunctions 261
8.1 Infiniteseries 261
8.2 Powerseries 267
8.3 HigherderivativesandTaylorpolynomials 275
8.4 Differentiationandintegrationofsequencesoffunctions 281
8.5 Theexponentialfunction 287
8.6 Compactsubsetsin𝐶[𝑎,𝑏] 290
8.7 Exercises 295
9 ChaosinDiscreteDynamicalSystems 301
9.1 Thedefinitionofchaos 302
9.2 Semiconjugacy 310
9.3 Subshiftsoffinitetype 314
9.4 Itinerariesandpiecewiseexpandingmaps 319
9.5 Adynamicalsystemwithadenseorbitbutnoperiodicpoints 331
9.6 Exercises 337