Continuous Nowhere Differentiable Functions: The Monsters of Analysis

Description

This book covers the construction, analysis, and theory of continuous nowhere differentiable functions, comprehensively and accessibly. After illuminating the significance of the subject through an overview of its history, the reader is introduced to the sophisticated toolkit of ideas and tricks used to study the explicit continuous nowhere differentiable functions of Weierstrass, Takagi–van der Waerden, Bolzano, and others. Modern tools of functional analysis, measure theory, and Fourier analysis are applied to examine the generic nature of continuous nowhere differentiable functions, as well as linear structures within the (nonlinear) space of continuous nowhere differentiable functions. To round out the presentation, advanced techniques from several areas of mathematics are brought together to give a state-of-the-art analysis of Riemann’s continuous, and purportedly nowhere differentiable, function. For the reader’s benefit, claims requiring elaboration, and open problems, are clearly indicated. An appendix conveniently provides background material from analysis and number theory, and comprehensive indices of symbols, problems, and figures enhance the book’s utility as a reference work. Students and researchers of analysis will value this unique book as a self-contained guide to the subject and its methods Front Matter....Pages i-xii Introduction: A Historical Journey....Pages 1-6 Front Matter....Pages 7-7 Preliminaries....Pages 9-17 Weierstrass-Type Functions I....Pages 19-49 Takagi–van der Waerden-Type Functions I....Pages 51-63 Bolzano-Type Functions I....Pages 65-97 Other Examples....Pages 99-104 Front Matter....Pages 105-105 Baire Category Approach....Pages 107-130 Front Matter....Pages 131-131 Weierstrass-Type Functions II....Pages 133-186 Takagi–van der Waerden-Type Functions II....Pages 187-201 Bolzano-Type Functions II....Pages 203-208 Besicovitch Functions....Pages 209-244 Linear Spaces of Nowhere Differentiable Functions....Pages 245-254 Front Matter....Pages 255-255 Riemann Function....Pages 257-264 Back Matter....Pages 265-299