Knot points of typical continuous functions
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- by David Preiss and Shingo Saito PDF
- Trans. Amer. Math. Soc. 366 (2014), 833-856 Request permission
Abstract:
It is well known that most continuous functions are nowhere differentiable. Furthermore, in terms of Dini derivatives, most continuous functions are nondifferentiable in the strongest possible sense except in a small set of points. In this paper, we completely characterise families $\mathcal {S}$ of sets of points for which most continuous functions have the property that such small set of points belongs to $\mathcal {S}$. The proof uses a topological zero-one law and the Banach-Mazur game.References
- Stefan Banach, Über die Bairesche Kategorie gewisser Funktionenmengen, Studia Math. 3 (1931), 174–179.
- Vojtěch Jarník, Über die Differenzierbarkeit stetiger Funktionen, Fund. Math. 21 (1933), 48–58.
- Alexander S. Kechris, Classical descriptive set theory, Graduate Texts in Mathematics, vol. 156, Springer-Verlag, New York, 1995. MR 1321597, DOI 10.1007/978-1-4612-4190-4
- Stefan Mazurkiewicz, Sur les fonctions non dérivables, Studia Math. 3 (1931), 92–94.
- John C. Oxtoby, The Banach-Mazur game and Banach category theorem, Contributions to the theory of games, vol. 3, Annals of Mathematics Studies, no. 39, Princeton University Press, Princeton, N.J., 1957, pp. 159–163. MR 0093741
- David Preiss and Luděk Zajíček, On the differentiability structure of typical continuous functions, unpublished work.
- Shingo Saito, Residuality of families of $\scr F_\sigma$ sets, Real Anal. Exchange 31 (2005/06), no. 2, 477–487. MR 2265789, DOI 10.14321/realanalexch.31.2.0477
- Shingo Saito, Knot points of typical continuous functions and Baire category in families of sets of the first class, Ph.D. thesis, University of London, 2008.
- L. Zajíček, Differentiability properties of typical continuous functions, Real Anal. Exchange 25 (1999/00), no. 1, 149–158.
Additional Information
- David Preiss
- Affiliation: Mathematics Institute, University of Warwick, Coventry CV4 7AL, United Kingdom
- MR Author ID: 141890
- Email: d.preiss@warwick.ac.uk
- Shingo Saito
- Affiliation: Institute of Mathematics for Industry, Kyushu University, 744, Motooka, Nishi-ku, Fukuoka, 819-0395, Japan
- Address at time of publication: Faculty of Arts and Science, Kyushu University, 744, Motooka, Nishi-ku, Fukuoka, 819-0395, Japan
- MR Author ID: 783465
- Email: ssaito@imi.kyushu-u.ac.jp, ssaito@artsci.kyushu-u.ac.jp
- Received by editor(s): April 13, 2012
- Published electronically: September 26, 2013
- © Copyright 2013
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Trans. Amer. Math. Soc. 366 (2014), 833-856
- MSC (2010): Primary 26A27; Secondary 26A21, 28A05, 54H05
- DOI: https://doi.org/10.1090/S0002-9947-2013-06100-4
- MathSciNet review: 3130318