Strange Functions in Real Analysis

DescriptionStrange 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 ContentsIntroduction: Basic ConceptsCantor and Peano type functionsFunctions of first Baire classSemicontinuous functions that are not countably continuousSingular monotone functionsA characterization of constant functions via Dini’s derived numbersEverywhere differentiable nowhere monotone functionsContinuous nowhere approximately differentiable functionsBlumberg’s theorem and Sierpinski-Zygmund functionsThe cardinality of first Baire classLebesgue nonmeasurable functions and functions without the Baire propertyHamel basis and Cauchy functional equationSummation methods and Lebesgue nonmeasurable functionsLuzin sets, Sierpi´nski sets, and their applicationsAbsolutely nonmeasurable additive functionsEgorov type theoremsA difference between the Riemann and Lebesgue iterated integralsSierpinski’s partition of the Euclidean planeBad functions defined on second category setsSup-measurable and weakly sup-measurable functionsGeneralized step-functions and superposition operatorsOrdinary differential equations with bad right-hand sidesNondifferentiable functions from the point of view of category and measureAbsolute null subsets of the plane with bad orthogonal projectionsAppendix 1: Luzin’s theorem on the existence of primitivesAppendix 2: Banach limits on the real lineAuthor DescriptionProf. 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)