The prevalence of continuous nowhere differentiable functions. (English) Zbl 0861.26003

Summary: In the space of continuous functions of a real variable, the set of nowhere differentiable functions has long been known to be topologically “generic”. In this paper it is shown further that in a measure theoretic sense (which is different from Wiener measure), “almost every” continuous function is nowhere differentiable. Similar results concerning other types of regularity, such as Hölder continuity, are discussed.


26A27 Nondifferentiability (nondifferentiable functions, points of nondifferentiability), discontinuous derivatives
28C20 Set functions and measures and integrals in infinite-dimensional spaces (Wiener measure, Gaussian measure, etc.)
26A21 Classification of real functions; Baire classification of sets and functions
60B11 Probability theory on linear topological spaces
26A16 Lipschitz (Hölder) classes
Full Text: DOI


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