The prevalence of continuous nowhere differentiable functions
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- by Brian R. Hunt PDF
- Proc. Amer. Math. Soc. 122 (1994), 711-717 Request permission
Abstract:
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.References
-
H. Auerbach and S. Banach, Uber die Höldersche Bedingung, Studia Math. 3 (1931), 180-184.
S. Banach, Über die Baire’sche Kategorie gewisser Funktionenmengen, Studia Math. 3 (1931), 174-179.
- Jens Peter Reus Christensen, On sets of Haar measure zero in abelian Polish groups, Israel J. Math. 13 (1972), 255–260 (1973). MR 326293, DOI 10.1007/BF02762799
- G. H. Hardy, Weierstrass’s non-differentiable function, Trans. Amer. Math. Soc. 17 (1916), no. 3, 301–325. MR 1501044, DOI 10.1090/S0002-9947-1916-1501044-1
- 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. MR 1161274, DOI 10.1090/S0273-0979-1992-00328-2
- 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. MR 1191479, DOI 10.1090/S0273-0979-1993-00396-3
- 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. MR 766105, DOI 10.1017/S0143385700002431
- K. Kuratowski, Topology. Vol. I, Academic Press, New York-London; Państwowe Wydawnictwo Naukowe [Polish Scientific Publishers], Warsaw, 1966. New edition, revised and augmented; Translated from the French by J. Jaworowski. MR 0217751
- R. Daniel Mauldin, The set of continuous nowhere differentiable functions, Pacific J. Math. 83 (1979), no. 1, 199–205. MR 555048
- 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. MR 815038 S. Mazurkiewicz, Sur les fonctions non dérivables, Studia Math. 3 (1931), 92-94.
- 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. MR 584443 D. L. Renfro, Some supertypical nowhere differentiability results for $C[0,1]$, Doctoral Dissertation, North Carolina State University, 1993.
Additional Information
- © Copyright 1994 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 122 (1994), 711-717
- MSC: Primary 26A27; Secondary 26A16, 28C20, 60B11
- DOI: https://doi.org/10.1090/S0002-9939-1994-1260170-X
- MathSciNet review: 1260170