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

Kolář, Jan

doi:10.1090/S0002-9939-00-05811-1

Porous sets that are Haar null, and nowhere approximately differentiable functions
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Proc. Amer. Math. Soc. 129 (2001), 1403-1408 Request permission

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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