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On differentiability properties of typical continuous functions and Haar null sets

Author: L. Zajícek
Journal: Proc. Amer. Math. Soc. 134 (2006), 1143-1151
MSC (2000): Primary 26A27; Secondary 28C20
Published electronically: September 28, 2005
MathSciNet review: 2196050
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Abstract: Let $ D$ ($ D^*$) be the set of all continuous functions $ f$ on $ [0,1]$ which have a derivative $ f'(x)\in \mathbf{R}$ ( $ f'(x)\in \mathbf{R}^*$, respectively) at least at one point $ x \in (0,1)$. B. R. Hunt (1994) proved that $ D$ is Haar null (in Christensen's sense) in $ C[0,1]$.

In the present article it is proved that neither $ D^*$ nor its complement is Haar null in $ C[0,1]$. Moreover, the same assertion holds if we consider the approximate derivative (or the ``strong'' preponderant derivative) instead of the ordinary derivative; these results are proved using a new result on typical (in the sense of category) continuous functions, which is of interest in its own right.

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Additional Information

L. Zajícek
Affiliation: Charles University, Faculty of Mathematics and Physics, Sokolovská 83, 186 75 Prague 8, Czech Republic

Keywords: Typical continuous function, Haar null set, nowhere differentiable function, approximative derivative, preponderant derivative
Received by editor(s): March 5, 2004
Received by editor(s) in revised form: November 9, 2004
Published electronically: September 28, 2005
Additional Notes: This research was supported by MSM 113200007, GAČR 201/00/0767 and GAČR 201/03/0931
Communicated by: David Preiss
Article copyright: © Copyright 2005 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.