Synopses & Reviews
Weierstrass and Blancmange nowhere differentiable functions, Lebesgue integrable functions with everywhere divergent Fourier series, and various nonintegrable Lebesgue measurable functions. While dubbed strange or pathological, these functions are ubiquitous throughout mathematics and play an important role in analysis, not only as counterexamples of seemingly true and natural statements, but also to stimulate and inspire the further development of real analysis.
Strange Functions in Real Analysis explores a number of important examples and constructions of pathological functions. After introducing the basic concepts, the author begins with Cantor and Peano-type functions, then moves to functions whose constructions require essentially noneffective 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, he considers examples of functions whose existence cannot be established without the help of additional set-theoretical axioms and demonstrates that their existence follows from certain set-theoretical hypotheses, such as the Continuum Hypothesis.
Synopsis
This book explores strange functions in real analysis and their applications. This book presents basic set-theoretical concepts such as binary relations of special type and the Generalized Continuum Hypothesis. It examines various functions whose constructions need essentially noneffective methods and those whose existence arises from known hypotheses. It also discusses basic concepts of general topology and classical descriptive set theory. This second edition features five new chapters, with revised material throughout the text. It includes additional exercises as well as an expanded reference list. Strange Functions in Real Analysis is a valuable resource for students and mathematicians.