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.