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Surprises and Counterexamples in Real Function Theory

A. R. Rajwade and A. K. Bhandari
Hindustan Book Agency
Publication Date: 
Number of Pages: 
Texts and Readings in Mathematics 42
[Reviewed by
Kenneth A. Ross
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The preface of the book describes the book very well. After the following excerpts, I will add my own observations.

Our aim in this book is to consider a variety of intriguing, surprising, and appealing topics, and non-routine 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 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…

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. For example: We know that continuous functions defined on an interval satisfy the intermediate value property. Are there functions which are not continuous but have this property in every interval? If a one-to-one function is continuous at a point, is its inverse also continuous at the image of that point? Most would believe that there is a tangent to the graph of a function if and only if the function is differentiable at the point, but a counter-example is given. Where does one find easily accessible details of everywhere continuous, nowhere differentiable functions? (Chapter 3)

In addition to standard topics, a few of which are mentioned above, many extra topics are presented in detail, including periodic functions, the Mean Value Theorem, inflection points, harmonic series, and Schwarz differentiability (in an appendix). Not much is overlooked, but I would have liked a section on upper and lower semi-continuous functions.

This fine book contains classical concrete treatments. The authors are clearly fearless analysts, who don’t shun technical details. American students might find the going tough, though the exposition is careful and detailed.

Some American readers may find the treatments old-fashioned. Modern ideas are avoided. For example, Chapter 3 is devoted to constructing continuous non-differentiable functions, but there’s no mention that “most” continuous functions are non-differentiable thanks to the Baire Category Theorem.

The text is very clean; I found about five misprints. The index is surprisingly short and incomplete, considering the careful preparation of the rest of the book. I am impressed with the extremely complete set of references. So far as I can tell, they are all referred to in sufficient detail to make the text historically very interesting. Every relevant reference that came to my mind was already there! It should be noted that, except for two books, all 118 references were published prior to 1991.

Teachers of the theory of calculus will benefit by having this book on their shelves; I wish I’d had it available when I was teaching. See also the book Strange Functions in Real Analysis by A. B. Kharazishvili.

Kenneth A. Ross ( taught at the University of Oregon from 1965 to 2000. His research area of interest was commutative harmonic analysis, especially where it has a probabilistic flavor.

  • 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 result
  • The infinite exponential $x\thinsp x\thinsp x\thinsp :\thinsp :\thinsp :$ 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