Continuous Nowhere Differentiable Functions

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.

“This book is a thorough survey of a function that, at its announcement, took the mathematical world by storm. This book can be recommended for those whose research involves working in analysis. … Of interest to many will be the open problems placed throughout the text, along with the extensive bibliography. … a very detailed book which, if you work in classical analysis or topology or expect to work with students interested in this topic, belongs in your collection.” (Robert W. Vallin, Mathematical Reviews, June, 2016)

“By bringing together results scattered in various publications, some of them hardly to find or/and hardly to read (I mean old papers), presenting them in a unitary and rigorous way (using a modern language and style) with pertinent historical comments, the authors have done a great service to the mathematical community. The book presents interest for all mathematicians, but also for people (engineers, physicists, etc) having a basic background in calculus, interested in the evolution … .” (S. Cobzaş, Studia Universitatis Babes-Bolyai, Mathematica, Vol. 61 (1), 2016)

“The book presents the construction, analysis, and theory of continuous nowhere differentiable functions in a comprehensive and accessible manner. … This unique book will be of interest to students and researchers of analysis as a self-contained guide to the subject of continuous nowhere differentiable functions and its methods.” (Zoltán Finta, zbMATH 1334.26001, 2016)

Marek Jarnicki is Professor of Mathematics at Jagiellonian University, Poland. His primary subject of research is complex analysis, particularly holomorphically invariant (contractible) pseudodistances and pseudometrics; domains of holomorphy with respect to special cases of holomorphic functions; continuation of holomorphic functions with restricted growth; and the extension of separately analytic functions.

Peter Pflug is Professor of Mathematics at the University of Oldenburg, Germany. His primary subject of research is the theory of functions of several complex variables and complex analysis.