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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.

MSC:

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
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