3rd Edition

Strange Functions in Real Analysis

By Alexander Kharazishvili Copyright 2018
    440 Pages 25 B/W Illustrations
    by Chapman & Hall

    440 Pages 25 B/W Illustrations
    by Chapman & Hall

    Strange Functions in Real Analysis, Third Edition differs from the previous editions in that it includes five new chapters as well as two appendices. More importantly, the entire text has been revised and contains more detailed explanations of the presented material. In doing so, the book explores a number of important examples and constructions of pathological functions.

    After introducing basic concepts, the author begins with Cantor and Peano-type functions, then moves effortlessly to functions whose constructions require what is essentially non-effective methods. These include functions without the Baire property, functions associated with a Hamel basis of the real line and Sierpinski-Zygmund functions that are discontinuous on each subset of the real line having the cardinality continuum.

    Finally, the author considers examples of functions whose existence cannot be established without the help of additional set-theoretical axioms. On the whole, the book is devoted to strange functions (and point sets) in real analysis and their applications.

    Introduction: basic concepts

    Cantor and Peano type functions

    Functions of first Baire class

    Semicontinuous functions that are not countably continuous

    Singular monotone functions

    A characterization of constant functions via Dini’s derived numbers

    Everywhere differentiable nowhere monotone functions

    Continuous nowhere approximately differentiable functions

    Blumberg’s theorem and Sierpinski-Zygmund functions

    The cardinality of first Baire class

    Lebesgue nonmeasurable functions and functions without the Baire property

    Hamel basis and Cauchy functional equation

    Summation methods and Lebesgue nonmeasurable functions

    Luzin sets, Sierpi´nski sets, and their applications

    Absolutely nonmeasurable additive functions

    Egorov type theorems

    A difference between the Riemann and Lebesgue iterated integrals

    Sierpinski’s partition of the Euclidean plane

    Bad functions defined on second category sets

    Sup-measurable and weakly sup-measurable functions

    Generalized step-functions and superposition operators

    Ordinary differential equations with bad right-hand sides

    Nondifferentiable functions from the point of view of category and measure

    Absolute null subsets of the plane with bad orthogonal projections

    Appendix 1: Luzin’s theorem on the existence of primitives

    Appendix 2: Banach limits on the real line



    Prof. A. Kharazishvili is Professor I. Chavachavadze Tibilisi State University, an author of more than 200 scientific works in various branches of mathematics (set theory, combinatorics and graph theory, mathematical analysis, convex geometry and probability theory). He is an author of several monographs. The author is a member of the Editorial Board of Georgian Mathematical Journal (Heldermann-Verlag), Journal of Applied Analysis (Heldermann-Verlag), Journal of Applied Mathematics, Informatics and Mechanics (Tbilisi State University Press)

    This is the third edition of a text based on the author's lectures at Tiblisi University, Georgia. While of interest in themselves, the "strange functions" alluded to in the title can serve as counterexamples to hypotheses that on first consideration appear reasonable. Thus, they inform mathematical thinking in the field. The text also provides the mathematical framework used to develop and validate these strange functions. Other reviewers of past editions of this book have observed that it is similar in concept to J. C. Oxtoby's Measure and Category (1971). This edition contains more examples and is substantially longer than Oxtoby's. Kharazishvili has added five chapters and two appendixes to the second edition (2005) and presents a fairly complete revision of that edition. While this work is as much a reference as it is a textbook, it contains a number of exercises as well as an extensive bibliography. This text is recommended for advanced mathematics collections, though there may not be sufficient new material to justify replacing the previous edition.

    --D. Z. Spicer, University System of Maryland, Choice Connect