Texts and Readings in Mathematics

This book presents a variety of intriguing, surprising and appealing topics and nonroutine proofs of several theorems in real function theory. It is a reference book to which one can turn for finding answers to curiosities that arise while studying or teaching analysis.

Chapter 1 is an introduction to algebraic, irrational and transcendental numbers and contains the construction of the Cantor ternary set. Chapter 2 contains functions with extraordinary properties. Chapter 3 discusses functions that are continuous at each point but differentiable at no point. Chapters 4 and 5 include the intermediate value property, periodic functions, Rolle's theorem, Taylor's theorem, points of inflexion and tangents. Chapter 6 discusses sequences and series. It includes the restricted harmonic series, rearrangements of alternating harmonic series and some number theoretic aspects. In Chapter 7, the infinite exponential with its peculiar range of convergence is studied. Appendix I deal with some specialized topics. Exercises are included at the end of chapters and their solutions are provided in Appendix II.
1. Introduction to the real line R and some of its subsets
2. Functions: Pathological, peculiar and extraordinary
3. Famous everywhere continuous, nowhere differentiable functions: van der Waerden's and others
4. Functions: Continuous, periodic, locally recurrent and others
5. The derivative and higher derivatives
6. Sequences, Harmonic Series, Alternating Series and related result
7. The infinite exponential and related results
A.1. Stirling's formula and the trapezoidal rule
A.2. Schwarz differentiability
A.3. Cauchy's functional equation f(x + y) = f(x) + f(y)
Appendix II: Hints and solutions to exercises