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


[1] H. Auerbach and S. Banach, Uber die Höldersche Bedingung, Studia Math. 3 (1931), 180-184. · JFM 57.0306.01
[2] S. Banach, Über die Baire’sche Kategorie gewisser Funktionenmengen, Studia Math. 3 (1931), 174-179. · JFM 57.0305.05
[3] Jens Peter Reus Christensen, On sets of Haar measure zero in abelian Polish groups, Proceedings of the International Symposium on Partial Differential Equations and the Geometry of Normed Linear Spaces (Jerusalem, 1972), 1972, pp. 255 – 260 (1973). · Zbl 0249.43002
[4] G. H. Hardy, Weierstrass’s non-differentiable function, Trans. Amer. Math. Soc. 17 (1916), no. 3, 301 – 325. · JFM 46.0401.03
[5] Brian R. Hunt, Tim Sauer, and James A. Yorke, Prevalence: a translation-invariant ”almost every” on infinite-dimensional spaces, Bull. Amer. Math. Soc. (N.S.) 27 (1992), no. 2, 217 – 238. · Zbl 0763.28009
[6] Brian R. Hunt, Tim Sauer, and James A. Yorke, Prevalence. An addendum to: ”Prevalence: a translation-invariant ’almost every’ on infinite-dimensional spaces” [Bull. Amer. Math. Soc. (N.S.) 27 (1992), no. 2, 217 – 238; MR1161274 (93k:28018)], Bull. Amer. Math. Soc. (N.S.) 28 (1993), no. 2, 306 – 307. · Zbl 0763.28009
[7] James L. Kaplan, John Mallet-Paret, and James A. Yorke, The Lyapunov dimension of a nowhere differentiable attracting torus, Ergodic Theory Dynam. Systems 4 (1984), no. 2, 261 – 281. · Zbl 0558.58018
[8] K. Kuratowski, Topology. Vol. I, New edition, revised and augmented. Translated from the French by J. Jaworowski, Academic Press, New York-London; Państwowe Wydawnictwo Naukowe, Warsaw, 1966. · Zbl 0158.40901
[9] R. Daniel Mauldin, The set of continuous nowhere differentiable functions, Pacific J. Math. 83 (1979), no. 1, 199 – 205. · Zbl 0387.46025
[10] R. Daniel Mauldin, Correction: ”The set of continuous nowhere differentiable functions” [Pacific J. Math. 83 (1979), no. 1, 199 – 205; MR0555048 (81g:46033)], Pacific J. Math. 121 (1986), no. 1, 119 – 120. · Zbl 0387.46025
[11] S. Mazurkiewicz, Sur les fonctions non dérivables, Studia Math. 3 (1931), 92-94. · Zbl 0003.29702
[12] John C. Oxtoby, Measure and category, 2nd ed., Graduate Texts in Mathematics, vol. 2, Springer-Verlag, New York-Berlin, 1980. A survey of the analogies between topological and measure spaces. · Zbl 0435.28011
[13] D. L. Renfro, Some supertypical nowhere differentiability results for \( C[0,1]\), Doctoral Dissertation, North Carolina State University, 1993.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.