Problems in Mathematical Analysis II: Continuity and Differentiation

418 Pages·2001·2.049 MB·English

by  W. J. Kaczor, M. T. Nowak

STUDENT MATHEMATICAL LIBRARY
Volume 12
Problems in
Mathematical
Analysis II
Continuity and
Differentiation
W. J. Kaczor
M. T. Nowak
Selected Titles in This Series
12 W. J. Kaczor and M. T. Nowak, Problemsinmathematicalanalysis
II:Continuityanddifferentiation,2001
11 Michael Mesterton-Gibbons, Anintroductiontogame-theoretic
modelling,2000
10 John Oprea, Themathematicsofsoapfilms:ExplorationswithMaple(cid:2)r,
2000
9 David E. Blair, Inversiontheoryandconformalmapping,2000
8 Edward B. Burger, Exploringthenumberjungle:Ajourneyinto
diophantineanalysis,2000
7 Judy L. Walker, Codesandcurves,2000
6 G´erald Tenenbaum and Michel Mend`es France, Theprimenumbers
andtheirdistribution,2000
5 Alexander Mehlmann, Thegame’safoot!Gametheoryinmythand
paradox,2000
4 W. J. Kaczor and M. T. Nowak, Problemsinmathematicalanalysis
I:Realnumbers,sequencesandseries,2000
3 Roger Knobel, Anintroductiontothemathematicaltheoryofwaves,
2000
2 Gregory F. Lawler and Lester N. Coyle, Lecturesoncontemporary
probability,1999
1 Charles Radin, Milesoftiles,1999

Problems in
Mathematical
Analysis II
Continuity and
Differentiation

STUDENT MATHEMATICAL LIBRARY
Volume 12
Problems in
Mathematical
Analysis II
Continuity and
Differentiation
W. J. Kaczor
M. T. Nowak
Editorial Board
David Bressoud Carl Pomerance
Robert Devaney, Chair Hung-Hsi Wu
Originallypublished in Polish,as
Zadania z Analizy Matematycznej. Cze(cid:2)´s´c Druga
Funkcje Jednej Zmiennej—Rachunek R´oz˙niczowy
(cid:2)c 1998, Wydawnictwo Uniwersytetu Marii Curie–Sk(cid:2)lodowskiej, Lublin.
Translated, revised and augmented by the authors.
2000 Mathematics Subject Classification. Primary00A07;
Secondary 26A06, 26A15, 26A24.
Library of Congress Cataloging-in-Publication Data
Kaczor,W.J.(WieslawaJ.),1949–
[Zadaniazanalizymatematycznej. English]
Problems in mathematical analysis. I. Real numbers, sequences and series /
W.J.Kaczor,M.T.Nowak.
p.cm. —(Studentmathematicallibrary,ISSN1520-9121;v.4)
Includesbibliographicalreferences.
ISBN0-8218-2050-8(softcover;alk.paper)
1. Mathematical analysis I. Nowak, M. T. (Maria T.), 1951– II. Title.
III.Series.
QA300K32513 2000
515(cid:2).076–dc21 99-087039
Copying and reprinting. Individual readers of this publication, and nonprofit
libraries acting for them, are permitted to make fair use of the material, such as to
copy a chapter for use in teaching or research. Permission is granted to quote brief
passagesfromthispublicationinreviews,providedthecustomaryacknowledgmentof
thesourceisgiven.
Republication,systematiccopying,ormultiplereproductionofanymaterialinthis
publicationispermittedonlyunderlicensefromtheAmericanMathematicalSociety.
Requests for such permission should be addressed to the Assistant to the Publisher,
AmericanMathematicalSociety,P.O.Box6248,Providence,RhodeIsland02940-6248.
[email protected].
(cid:2)c 2001bytheAmericanMathematicalSociety. Allrightsreserved.
TheAmericanMathematicalSocietyretainsallrights
exceptthosegrantedtotheUnitedStatesGovernment.
PrintedintheUnitedStatesofAmerica.
(cid:2)∞ Thepaperusedinthisbookisacid-freeandfallswithintheguidelines
establishedtoensurepermanenceanddurability.
VisittheAMShomepageatURL:http://www.ams.org/
10987654321 060504030201
Contents
Preface xi
Notation and Terminology xiii
Problems
Chapter (cid:2)(cid:3) Limits and Continuity
(cid:2)(cid:3)(cid:2)(cid:3) The Limit of a Function (cid:4)
(cid:2)(cid:3)(cid:5)(cid:3) Properties of Continuous Functions (cid:6)
(cid:2)(cid:3)(cid:4)(cid:3) Intermediate Value Property (cid:2)(cid:7)
(cid:2)(cid:3)(cid:7)(cid:3) Semicontinuous Functions (cid:2)(cid:8)
(cid:2)(cid:3)(cid:9)(cid:3) Uniform Continuity (cid:5)(cid:7)
(cid:2)(cid:3)(cid:10)(cid:3) Functional Equations (cid:5)(cid:11)
(cid:2)(cid:3)(cid:11)(cid:3) Continuous Functions in Metric Spaces (cid:4)(cid:5)
Chapter (cid:5)(cid:3) Di(cid:12)erentiation
(cid:5)(cid:3)(cid:2)(cid:3) The Derivative of a Real Function (cid:4)(cid:11)
(cid:5)(cid:3)(cid:5)(cid:3) Mean Value Theorems (cid:7)(cid:9)
(cid:5)(cid:3)(cid:4)(cid:3) Taylor(cid:13)s Formula and L(cid:13)Hospital(cid:13)s Rule (cid:9)(cid:5)
(cid:5)(cid:3)(cid:7)(cid:3) Convex Functions (cid:10)(cid:2)
vii
viii Contents
(cid:5)(cid:3)(cid:9)(cid:3) Applications of Derivatives (cid:10)(cid:8)
(cid:5)(cid:3)(cid:10)(cid:3) StrongDi(cid:12)erentiabilityandSchwarzDi(cid:12)erentiability (cid:11)(cid:11)
Chapter (cid:4)(cid:3) Sequences and Series of Functions
(cid:4)(cid:3)(cid:2)(cid:3) Sequences of Functions(cid:14) Uniform Convergence (cid:8)(cid:2)
(cid:4)(cid:3)(cid:5)(cid:3) Series of Functions(cid:14) Uniform Convergence (cid:8)(cid:11)
(cid:4)(cid:3)(cid:4)(cid:3) PowerSeries (cid:6)(cid:10)
(cid:4)(cid:3)(cid:7)(cid:3) Taylor Series (cid:2)(cid:15)(cid:5)
Solutions
Chapter (cid:2)(cid:3) Limits and Continuity
(cid:2)(cid:3)(cid:2)(cid:3) The Limit of a Function (cid:2)(cid:2)(cid:2)
(cid:2)(cid:3)(cid:5)(cid:3) Properties of Continuous Functions (cid:2)(cid:5)(cid:6)
(cid:2)(cid:3)(cid:4)(cid:3) Intermediate Value Property (cid:2)(cid:7)(cid:10)
(cid:2)(cid:3)(cid:7)(cid:3) Semicontinuous Functions (cid:2)(cid:10)(cid:15)
(cid:2)(cid:3)(cid:9)(cid:3) Uniform Continuity (cid:2)(cid:11)(cid:2)
(cid:2)(cid:3)(cid:10)(cid:3) Functional Equations (cid:2)(cid:8)(cid:2)
(cid:2)(cid:3)(cid:11)(cid:3) Continuous Functions in Metric Spaces (cid:2)(cid:6)(cid:8)
Chapter (cid:5)(cid:3) Di(cid:12)erentiation
(cid:5)(cid:3)(cid:2)(cid:3) The Derivative of a Real Function (cid:5)(cid:2)(cid:2)
(cid:5)(cid:3)(cid:5)(cid:3) Mean Value Theorems (cid:5)(cid:4)(cid:4)
(cid:5)(cid:3)(cid:4)(cid:3) Taylor(cid:13)s Formula and L(cid:13)Hospital(cid:13)s Rule (cid:5)(cid:7)(cid:9)
(cid:5)(cid:3)(cid:7)(cid:3) Convex Functions (cid:5)(cid:10)(cid:11)
(cid:5)(cid:3)(cid:9)(cid:3) Applications of Derivatives (cid:5)(cid:8)(cid:9)
(cid:5)(cid:3)(cid:10)(cid:3) StrongDi(cid:12)erentiabilityandSchwarzDi(cid:12)erentiability (cid:4)(cid:2)(cid:15)
Chapter (cid:4)(cid:3) Sequences and Series of Functions
(cid:4)(cid:3)(cid:2)(cid:3) Sequences of Functions(cid:14) Uniform Convergence (cid:4)(cid:2)(cid:11)
(cid:4)(cid:3)(cid:5)(cid:3) Series of Functions(cid:14) Uniform Convergence (cid:4)(cid:4)(cid:10)
(cid:4)(cid:3)(cid:4)(cid:3) PowerSeries (cid:4)(cid:9)(cid:9)
Contents ix
(cid:4)(cid:3)(cid:7)(cid:3) Taylor Series (cid:4)(cid:11)(cid:5)
Bibliography (cid:16) Books (cid:4)(cid:6)(cid:4)
Index (cid:4)(cid:6)(cid:11)