- (P)
- for every Darboux function there exists a continuous nowhere constant function such that is Darboux
- (A)
- for every subset of of cardinality there exists a uniformly continuous function such that ,
- (B)
- for an arbitrary function whose image contains a non-trivial interval there exists an of cardinality such that the restriction of to is uniformly continuous,
We define a new notion of ``HP-small' set which implies that is both -porous and Haar null in the sense of Christensen. We show that the set of all continuous functions on which have finite unilateral approximate derivative at a point 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 which are Hölder at a point is HP-small.
In the present article it is proved that neither nor its complement is Haar null in . 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.