The size of the Dini subdifferential

Abstract:

Given a lower semicontinuous function $f:\mathbb {R}^h \rightarrow \mathbb {R} \cup \{+\infty \}$, we prove that the points of $\mathbb {R}^h$, where the lower Dini subdifferential contains more than one element, lie in a countable union of sets which are isomorphic to graphs of some Lipschitzian functions defined on $\mathbb {R}^{h-1}$. Consequently, the set of all these points has a null Lebesgue measure.