Rajwade A.R., Bhandari A.K. Surprises and Counterexamples in Real Function Theory

New Delhi: Hindustan Book Agency, 2011. — 300 p. — (Texts and Readings in Mathematics 42) — ISBN: 9380250169.Our aim in this book is to consider a variety of intriguing, surprising and appealing topics, and nonroutine proofs of the usual results of real function theory. The reader is expected to have done a first course in real analysis (or advanced calculus), since the book assumes a knowledge of continuity and differentiability of functions, Rolle's theorem, the mean value theorem, Taylor expansion and Riemann integration. However, no sophisticated knowledge of analysis is required and a student at the masters or advanced undergraduate level should have no difficulty in going through the book. Though this book has some part of the title common with the book "Counterexamples in Analysis" by Gelbaum and Olmstead, it is totally different, in nature and contents. Some examples and counterexamples (fifteen or twenty) in our book are essentially the same as given in the <Abook by Gelbaum and Olmstead, but otherwise the intersection is small. This book contains a number of surprising and unexpected results. It is meant to be a reference book and is expected to be a book to which one turns for finding answers to curiosities which one comes across while studying or teaching elementary analysis.Introduction to the real line R and some of its subsets.
Functions: Pathological, peculiar and extraordinary.
Famous everywhere continuous, nowhere differentiable functions: van der Waerden's and others.
Functions: Continuous, periodic, locally recurrent and others.
The derivative and higher derivatives.
Sequences, Harmonic Series, Alternating Series and related results.
The infinite exponential results.
Appendix I.
Appendix II.
Hints and solutions to exercises.PDF scan (HQ)