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

Author(s): Jan Kolár
Journal: Proc. Amer. Math. Soc. 129 (2001), 1403-1408.
MSC (1991): Primary 26A27, 28C20, 26A16, 26A24
Posted: October 25, 2000
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Abstract:

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.

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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: 10.1090/S0002-9939-00-05811-1
PII: S 0002-9939(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
Posted: October 25, 2000
Additional Notes: The author was supported by the grants GAUK 165/99 and CEZ:J13/98:113200007.
Communicated by: David Preiss
Copyright of article: Copyright 2000, American Mathematical Society

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P. Holicky, L. Zajicek, Nondifferentiable functions, Haar null sets and Wiener measure, Acta Univ. Carolin. Math. Phys. 41 (2000), 7--11.

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