Strange Functions in Real Analysis
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.
Table of Contents
Introduction: Basic Concepts. Cantor and Peano Type Functions. Functions of First Baire Class (new). Semicontinuous Functions which are not Countably Continuous (new). Singular Monotone Functions. Everywhere Differentiable Nowhere Monotone Functions. Nowhere Approximately Differentiable Functions. Blumberg's Theorem and Sierpinski-Zygmund Function. 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.