On differentiability properties of typical continuous functions and Haar null sets
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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.References
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Additional Information
- L. Zajíček
- Affiliation: Charles University, Faculty of Mathematics and Physics, Sokolovská 83, 186 75 Prague 8, Czech Republic
- Email: zajicek@karlin.mff.cuni.cz
- 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
- © Copyright 2005
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Proc. Amer. Math. Soc. 134 (2006), 1143-1151
- MSC (2000): Primary 26A27; Secondary 28C20
- DOI: https://doi.org/10.1090/S0002-9939-05-08203-1
- MathSciNet review: 2196050