On Dini derivatives of real functions

Abstract

For a continuous function f , the set Vf made of those points where the lower left derivative is strictly less than the upper right derivative is totally disconnected. Besides continuity, alternative assumptions are proposed so to preserve this property. On the other hand, we construct a function f whose set Vf coincides with the entire domain, and nevertheless f is continuous on an infinite set, possibly having infinitely many cluster points. Some open problems are proposed.