Strange Functions in Real Analysis, second edition
This book presents a number of important examples and constructions of pathological sets and functions as well as their properties and applications. Although the author is interested in all types of strange sets and functions, the central topic of the monograph is the question of the existence of continuous functions that are not differentiable with respect to various concepts of generalized derivatives. The book is organized as follows. After recalling basic facts from set theory, topology and measure theory, examples of Cantor and Peano type functions are constructed. The author continues with basic properties of the space of Baire-one functions and with applications to separately continuous functions. Particular subclasses of semicontinuous functions are analysed in the next chapter. The next part is devoted to a study of the differentiability properties of real functions. Monotone functions are investigated and some pathological examples of monotone functions are constructed. The next chapters deal with everywhere differentiable nowhere monotone functions and nowhere approximately differentiable functions.
A connection between category and measurability is another important part of the monograph. Blumberg's theorem and Sierpinski-Zygmund functions are followed by examples and properties of Lebesgue nonmeasurable functions and functions without the Baire property. After that, bad solutions of the Cauchy functional equation are discussed. Luzin and Sierpinski sets along with applications are presented in the next chapter. Interesting relations between absolutely nonmeasurable functions and the measure extension problem are shown, followed by examples witnessing the fail of Egorov type theorems for pathological functions. Several results connected with Sierpinski's partition of the Euclidean plane follow. Examples of bad functions defined on second category sets conclude this part of the book. The next chapter is devoted to sup-measurable and weakly sup-measurable functions and to applications in the theory of ordinary differential equations.
In the final part of the book, the family of continuous nondifferentiable functions is considered from the point of view of category and measure. A short scheme for constructing the classical Wiener measure is presented along with some simple but useful statements from the theory of stochastic processes. Since the book is self-contained and the proofs are explained in detail, it should be accessible for all kinds of interested readers. All chapters are endowed with a large number of exercises that vary from elementary to difficult and provide a deeper insight into topics presented in the book. Thus the monograph is a good companion in the realm of pathological objects and counterexamples in real analysis.