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Strange Functions in Real Analysis

A. B. Kharazishvili
Publisher: 
Chapman & Hall/CRC
Publication Date: 
2006
Number of Pages: 
415
Format: 
Hardcover
Edition: 
2
Series: 
Pure and Applied Mathematics 272
Price: 
119.95
ISBN: 
1-58488-582-3
Category: 
Monograph
[Reviewed by
D. Benjamin Mathes
, on
06/13/2008
]

Strange Functions in Real Analysis is a good descriptive title for this book. It gives us numerous examples of behaviour reminiscent of such phenomena as everywhere continuous nowhere differentiablility.

The content has a significant intersection with Oxtoby's Measure and Category, but the exposition could not be more different. Oxtoby's book is as concise as a book could possibly be, and it is this quality that provides a thread through Oxtoby's exposition, giving the reader a glimpse of the big picture. For this reason I have used Oxtoby's book in our Analysis topics class, something I would not do with Kharazishvili's book.

Kharazishvili's book is big, and the thread is hard to discern, but being tangential in exposition to Oxtoby, it has qualities that are lacking in Oxtoby's book. Kharazishvili includes many more examples than Oxtoby does, and the book has numerous interesting exercises to try.


Ben Mathes is professor of mathematics at Colby College in Waterville, ME.

 Introduction: basic concepts
Cantor and Peano type functions
Functions of first Baire class New!
Semicontinuous functions that are not countably continuous New!
Singular monotone functions
Everywhere differentiable nowhere monotone functions
Nowhere approximately differentiable functions
Blumberg's theorem and Sierpinski-Zygmund functions
Lebesgue nonmeasurable functions and functions without the Baire property
Hamel basis and Cauchy functional equation
Luzin sets, Sierpinski sets, and their applications
Absolutely nonmeasurable additive functions New!
Egorov type theorems
Sierpinski's partition of the Euclidean plane
Bad functions defined on second category sets New!
Sup-measurable and weakly sup-measurable functions
Generalized step-functions and superposition operators New!
Ordinary differential equations with bad right-hand sides
Nondifferentiable functions from the point of view of category and measure
Bibliography
Subject Index

Dummy View - NOT TO BE DELETED