Chapman and Hall/CRC
426 pages | 25 B/W Illus.
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.
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
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