# A Primer of Real Functions

Cambridge University Press, 1996 - Mathematics - 305 pages
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.

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### Contents

 Sets 1 Sets of real numbers 5 Countable and uncountable sets 8 Metric spaces 21 Open and closed sets 25 Dense and nowhere dense sets 38 Compactness 45 Convergence and completeness 52
 Approximations to continuous functions 126 Linear functions 132 Derivatives 139 Monotonic functions 158 Convex functions 175 Infinitely differentiable functions 186 Integration 195 Measurable functions 201

 Nested sets and Baires theorem 61 Some applications of Baires Theorem 66 Sets of measure zero 73 Functions 77 Continuous functions 83 Properties of continuous functions 90 Upper and lower limits 105 Sequences of functions 108 Uniform convergence 112 Pointwise limits of continuous functions 123
 Definition of the Lebesgue integral 206 Properties of Lebesgue integrals 211 Applications of the Lebesgue integral 217 Stieltjes integrals 224 Applications of the Stieltjes integral 229 Partial sums of infinite series 237 Answers to Exercises 245 Index 281 Copyright