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

DOI:
https://doi.org/10.1090/S0002-9939-00-05811-1

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 which implies that is both -porous and Haar null in the sense of Christensen. We show that the set of all continuous functions on which have finite unilateral approximate derivative at a point 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 which are Hölder at a point is HP-small.

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Additional Information

**Jan Kolár**

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

Email:
kolar@karlin.mff.cuni.cz

DOI:
https://doi.org/10.1090/S0002-9939-00-05811-1

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