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Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.48.

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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
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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