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Let D (D*) be the set of all continuous functions f on [0,1] which have a derivative f'(x) ∈ R (f'(x) ∈ R*, respectively) at least at one point x ∈ (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.

On differentiability properties of typical continuous functions and Haar null sets