On differentiability properties of typical continuous functions and Haar null sets

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.