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Strange functions in real analysis. (English) Zbl 0942.26001
Pure and Applied Mathematics, Marcel Dekker. 229. New York, NY: Marcel Dekker. viii, 297 p. $ 175.00 (2000).

This book is devoted to strange (or singular) objects in various fields of mathematics, or, in other words, its “monsters and freaks”. Almost each mathematician “is acquainted” of them; here one can mention nowhere differentiable functions, nonmeasurable sets, measurable functions nonintegrable on any nonempty open subinterval, integrable functions with everywhere divergent Fourier series, strange curves covering sets of positive measure, and so on. Such “monsters and freaks” can be met in different fields in mathematics, numerous of them are collected, for example, in the well-known book by B. R. Gelbaum and J. M. H. Olmsted [“Theorems and counterexamples in analysis” (1990; Zbl 0702.00005)]. The present book is devoted only to “monsters and freaks” in elementary analysis where the meeting with them always is shocking. The author gives a systematic account of strange objects, explains connections and relations between them and axioms of set theory (although the author works in the framework of the “naive set theory”). His main attention is paid to real functions of real variables, but in general the book covers functions of more complicated structure. Mostly the approach to and point of view of the author at pathological objects are original and make transparent unexpected sides of phenomena, discover some unexpected connections and relations with other (and often usual) objects in mathematics. The author is well-known due to his studies in this field and the book is based on his course of lectures given at the Institute of Applied Mathematics of Tbilisi State University in 1998-1999 academic year.

The book consists of 13 small chapters: Introduction: basic concepts, Cantor and Peano type functions, Singular monotone functions, Everywhere differentiable nowhere monotone functions, Nowhere approximately differentiable functions, Blumberg’s theorem and Sierpiński-Zygmund function, Lebesgue nonmeasurable functions and functions without the Baire property, Hamel basis and Cauchy functional equations, Luzin sets, Sierpiński sets and their applications, Egorov type theorems, Sierpiński’s partition of the Euclidean plane, Sup-measurable and weakly sup-measurable functions, Ordinary differential equations with bad right-hand sides, Nondifferentiable functions from the point of view of category and measure. The Bibliography consists of 160 items.

One can see that this book is interesting for a wide circle of mathematicians who work in analysis and close branches of mathematics, as specialists as well as beginners. The book is written with clear and understandable language and is accessible even for graduate students.

26-02Research monographs (real functions)
26A27Nondifferentiability of functions of one real variable; discontinuous derivatives
26A30Real functions of one real variable with other special properties