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.
By:
Alexander Kharazishvili
Imprint: Chapman & Hall/CRC
Country of Publication: United States
Edition: 3rd edition
Dimensions:
Height: 234mm,
Width: 156mm,
Weight: 771g
ISBN: 9781498773140
ISBN 10: 1498773141
Pages: 440
Publication Date: 13 October 2017
Audience:
Professional and scholarly
,
College/higher education
,
Undergraduate
,
Primary
Format: Hardback
Publisher's Status: Active
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)
Reviews for Strange Functions in Real Analysis
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
