On Dini and Approximate Dini Derivates of Typical Continuous Functions

Abstract

In the thirties, Banach, Mazurkiewicz and Jarn\'{\i}k found relations connecting Dini derivates of a typical continuous function on $[0,1]$ at all points of $(0,1)$. We prove, answering a question of K. M. Garg, that there are no further relations of this sort. An analogous result is proved also for approximate Dini derivates. The aim of this note is to present relatively simple proofs of these results. An article containing an improvement of these results in several directions (with a considerably more complicated proof) is in preparation.