A Primer of Real Functions (Mathematical Association of America Textbooks, Series Number 13)

"This is a revised, updated, and augmented edition of a classic Carus monograph with a new chapter on integration and its applications. Earlier editions covered sets, metric spaces, continuous functions, and differentiable functions. To that, this edition adds sections on measurable sets and functions and the Lebesgue and Stieltjes integrals. The book retains the informal chatty style of the previous editions. It presents a variety of interesting topics, many of which are not commonly encountered in undergraduate textbooks, such as the existence of continuous everywhere-oscillating functions; two functions having equal derivatives, yet not differing by a constant; application of Stieltjes integration to the speed of convergence of infinite series. For readers with a background in calculus, the book is suitable either for self-study or for supplemental reading in a course on advanced calculus or real analysis. Students of mathematics will find here the sense of wonder that was associated with the subject in its early days"--Publisher description Front Cover......Page 1 A Primer of Real Functions......Page 5 Copyright Page......Page 4 Preface to the Fourth Edition......Page 9 Preface to the Third Edition......Page 11 Contents......Page 15 1 Sets......Page 17 2 Sets of real numbers......Page 21 3 Countable and uncountable sets......Page 24 4 Metric spaces......Page 37 5 Open and closed sets......Page 41 6 Dense and nowhere dense sets......Page 54 7 Compactness......Page 61 8 Convergence and completeness......Page 68 9 Nested sets and Baire's theorem......Page 77 10 Some applications of Baire's Theorem......Page 82 11 Sets of measure zero......Page 89 12 Functions......Page 93 13 Continuous functions......Page 99 14 Properties of continuous functions......Page 106 15 Upper and lower limits......Page 121 16 Sequences of functions......Page 124 17 Uniform convergence......Page 128 18 Pointwise limits of continuous functions......Page 139 19 Approximations to continuous functions......Page 142 20 Linear functions......Page 148 21 Derivatives......Page 155 22 Monotonic functions......Page 174 23 Convex functions......Page 191 24 Infinitely differentiable functions......Page 202 25 Lebesgue measure......Page 211 26 Measurable functions......Page 217 27 Definition of the Lebesgue integral......Page 222 28 Properties of Lebesgue integrals......Page 227 29 Applications of the Lebesgue integral......Page 233 30 Stieltjes integrals......Page 240 31 Applications of the Stieltjes integral......Page 245 32 Partial sums of infinite series......Page 253 Answers to Exercises......Page 261 Index......Page 297 Back Cover......Page 323 This is a revised, updated, and significantly augmented edition of a classic Carus Monograph (a bestseller for over 25 years) on the theory of functions of a real variable. Earlier editions of this classic Carus Monograph covered sets, metric spaces, continuous functions, and differentiable functions. The fourth edition adds sections on measurable sets and functions, the Lebesgue and Stieltjes integrals, and applications. The book retains the informal chatty style of the previous editions, remaining accessible to readers with some mathematical sophistication and a background in calculus. The book is, thus, suitable either for self-study or for supplemental reading in a course on advanced calculus or real analysis. Not intended as a systematic treatise, this book has more the character of a sequence of lectures on a variety of interesting topics connected with real functions. Many of these topics are not commonly encountered in undergraduate textbooks: e.g., the existence of continuous everywhere-oscillating functions (via the Baire category theorem); the universal chord theorem; two functions having equal derivatives, yet not differing by a constant; and application of Stieltjes integration to the speed of convergence of infinite series. This book recaptures the sense of wonder that was associated with the subject in its early days. It is a must for mathematics libraries. "This is a revised, updated, and significantly augmented edition of a classic Carus Monograph (a bestseller for over 25 years) on the theory of functions of a real variable. Earlier editions of this classic Carus Monograph covered sets, metric spaces, continuous functions, and differentiable functions. The fourth edition adds sections on measurable sets and functions, the Lebesgue and Stieltjes integrals, and applications. The book retains the informal chatty style of the previous editions, remaining accessible to readers with some mathematical sophistication and a background in calculus. The book is, thus, suitable either for self-study or for supplemental reading in a course on advanced calculus or real analysis. Not intended as a systematic treatise, this book has more the character of a sequence of lectures on a variety of interesting topics connected with real functions. Many of these topics are not commonly encountered in undergraduate textbooks: e.g., the existence of continuous everywhere-oscillating functions (via the Baire category theorem); the universal chord theorem; two functions having equal derivatives, yet not differing by a constant; and application of Stieltjes integration to the speed of convergence of infinite series."-- Publisher description This is a revised, updated and significantly augmented edition of a classic Carus Monograph (a bestseller for over 25 years) on the theory of functions of a real variable. Earlier editions of this classic Carus Monograph covered sets, metric spaces, continuous functions, and differentiable functions. The fourth edition adds sections on measurable sets and functions, the Lebesgue and Stieltjes integrals, and applications. The book retains the informal chatty style of the previous editions, remaining accessible to readers with some mathematical sophistication and a background in calculus. The book is thus suitable either for self-study or for supplemental reading in a course on advanced calculus or real analysis. Not intended as a systematic treatise, this book has more the character of a sequence of lectures on a variety of interesting topics connected with real functions. Many of these topics are not commonly encountered in undergraduate textbooks: for example, the existence of continuous everywhere-oscillating functions (via the Baire category theorem); the universal chord theorem; two functions having equal derivatives, yet not differing by a constant; and application of Stieltjes integration to the speed of convergence of infinite series

Revised edition of a classic Carus monograph with a new chapter on integration and its applications.

The Mathematics Teacher

I highly recommend this monograph to mathematics majors who wish to review or extend their knowledge of real analysis, and to workers in other fields who wish to learn of the major results of analysis.