Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



Porous sets that are Haar null, and nowhere approximately differentiable functions

Author: Jan Kolár
Journal: Proc. Amer. Math. Soc. 129 (2001), 1403-1408
MSC (1991): Primary 26A27, 28C20, 26A16, 26A24
Published electronically: October 25, 2000
MathSciNet review: 1814166
Full-text PDF

Abstract | References | Similar Articles | Additional Information


We define a new notion of ``HP-small'' set $A$ which implies that $A$ is both $\sigma$-porous and Haar null in the sense of Christensen. We show that the set of all continuous functions on $[0,1]$ which have finite unilateral approximate derivative at a point $x\in[0,1]$ is HP-small, as well as its projections onto hyperplanes. As a corollary, the same is true for the set of all Besicovitch functions. Also, the set of continuous functions on $[0,1]$ which are Hölder at a point is HP-small.

References [Enhancements On Off] (What's this?)

  • 1. V. Anisiu, Porosity and continuous, nowhere differentiable functions, Ann. Fac. Sci. Toulouse Math. (6) 2 (1993), 5-14. MR 94h:26007
  • 2. S. Banach, Über die Baire'sche Kategorie gewisser Funktionenmengen, Studia Math. 3 (1931), 174-179.
  • 3. A. Bruckner, Differentiation of Real Functions [2nd edition], CRM Monograph Series, Providence, Rhode Island, 1994. MR 94m:26001
  • 4. J. P. R. Christensen, Topology and Borel structure, North-Holland, Amsterdam, 1974. MR 50:1221
  • 5. P. M. Gandini and A. Zucco, Porosity and typical properties of real-valued continuous functions, Abh. Math. Sem. Univ. Hamburg 59 (1989), 15-22. MR 91c:26007
  • 6. B. R. Hunt, The prevalence of continuous nowhere differentiable functions, Proc. Amer. Math. Soc. 122 (1994), 711-717. MR 95d:26009
  • 7. E. Matousková, The Banach-Saks property and Haar null sets, Comment. Math. Univ. Carolinae 39 (1998), 71-80. MR 99g:46013
  • 8. S. Mazurkiewicz, Sur les fonctions non dérivables, Studia Math. 3 (1931), 92-94.
  • 9. D. L. Renfro, Some supertypical nowhere differentiability results for ${\mathcal C}[0,1]$, Doctoral Dissertation, North Carolina State University, 1993.
  • 10. L. Zajícek, Porosity and $\sigma$-porosity, Real Anal. Exchange 13 (1987-88), 314-350. MR 89e:26009

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (1991): 26A27, 28C20, 26A16, 26A24

Retrieve articles in all journals with MSC (1991): 26A27, 28C20, 26A16, 26A24

Additional Information

Jan Kolár
Affiliation: Department of Mathematical Analysis, Charles University, Sokolovská 83, 18675 Praha 8, Czech Republic

Keywords: Typical continuous function, $\sigma$-porous sets, Haar null sets, approximate derivative, Besicovitch functions, nowhere H\"older functions
Received by editor(s): August 9, 1999
Published electronically: October 25, 2000
Additional Notes: The author was supported by the grants GAUK 165/99 and CEZ:J13/98:113200007.
Communicated by: David Preiss
Article copyright: © Copyright 2000 American Mathematical Society

American Mathematical Society