The size of the Dini subdifferential

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.