On Dini derivatives of real functions

For a continuous function $f$, the set $V_f$ 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, for any given totally disconnected closed set $A$, we construct a function $f$ whose set $V_f$ coincides with the entire domain, and $f$ is continuous on $A$.