Strange Functions in Real Analysis (3^rd Ed.)

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)