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A Primer of Real Functions

A Primer of Real Functions

revised and updated by HAROLD P. BOAS
Volume: 13
Copyright Date: 1996
Edition: 4
Pages: 321
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  • Book Info
    A Primer of Real Functions
    Book 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. This book recaptures the sense of wonder that was associated with the subject in its early days. A must for your mathematics library.

    eISBN: 978-1-61444-013-0
    Subjects: Mathematics

Table of Contents

  1. Front Matter
    (pp. i-vi)
  2. Preface to the Fourth Edition
    (pp. vii-viii)
    Harold P. Boas
  3. Preface to the Third Edition
    (pp. ix-xii)
    Ralph P. Boas Jr.
  4. Table of Contents
    (pp. xiii-xiv)
  5. Chapter 1 Sets
    (pp. 1-76)

    In order to read anything about our subject, you will have to learn the language that is used in it. I have tried to keep the number of technical terms as small as possible, but there is a certain minimum vocabulary that is essential. Much of it consists of ordinary words used in special senses; this practice has both advantages and disadvantages, but has in any case to be endured since it is now too late to change the language completely. Much of the standard language is taken from the theory of sets, a subject with which we are not...

  6. Chapter 2 Functions
    (pp. 77-194)

    In elementary mathematics, it is customary to say thatyis a function ofxif, whenxis given,yis determined (uniquely; we are not concerned with “multiple-valued functions”). This is a good working definition and one that suffices for most practical purposes. However, we should realize that it does not define function,” although it does give a definite meaning to some phrases containing this word. (In a somewhat similar way, we are accustomed to attaching a definite meaning to the phrase “y→ ∞” even though ∞ by itself has no meaning.) However, it is interesting, and...

  7. Chapter 3 Integration
    (pp. 195-244)

    So far, we have mostly been dealing, directly or indirectly, with what used to be called differential calculus, and was clearly separated from integral calculus. The word “integral” has generally been used in two different senses. One is the process of undoing differentiation to find what have been variously described as antiderivatives, primitives, or indefinite integrals. This is the technique of finding explicit formulas for antiderivatives, an art which has lost much of its importance now that there are not only extensive tables of antiderivatives, but also computer programs that can find antiderivatives faster than people can find them by...

  8. Answers to Exercises
    (pp. 245-280)
  9. Index
    (pp. 281-305)