Strange Functions in Real Analysis

Description

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.

Table of Contents

1. Introduction: Basic Concepts
2. Cantor and Peano type functions
3. Functions of first Baire class
4. Semicontinuous functions that are not countably continuous
5. Singular monotone functions
6. A characterization of constant functions via Dini’s derived numbers
7. Everywhere differentiable nowhere monotone functions
8. Continuous nowhere approximately differentiable functions
9. Blumberg’s theorem and Sierpinski-Zygmund functions
10. The cardinality of first Baire class
11. Lebesgue nonmeasurable functions and functions without the Baire property
12. Hamel basis and Cauchy functional equation
13. Summation methods and Lebesgue nonmeasurable functions
14. Luzin sets, Sierpi´nski sets, and their applications
15. Absolutely nonmeasurable additive functions
16. Egorov type theorems
17. A difference between the Riemann and Lebesgue iterated integrals
18. Sierpinski’s partition of the Euclidean plane
19. Bad functions defined on second category sets
20. Sup-measurable and weakly sup-measurable functions
21. Generalized step-functions and superposition operators
22. Ordinary differential equations with bad right-hand sides
23. Nondifferentiable functions from the point of view of category and measure
24. Absolute null subsets of the plane with bad orthogonal projections
25. Appendix 1: Luzin’s theorem on the existence of primitives
26. Appendix 2: Banach limits on the real line

Author Description

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)