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This book explores a number of important examples and constructions of pathological functions. After introducing the basic concepts, the author begins with Cantor and Peano-type functions, then moves to functions whose constructions require essentially noneffective methods.
Table des matières
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
A propos de l'auteur
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)
Résumé
This book explores a number of important examples and constructions of pathological functions. After introducing the basic concepts, the author begins with Cantor and Peano-type functions, then moves to functions whose constructions require essentially noneffective methods.
