Given a locally Lipschitz continuous function $f: \mathbb{R}^{n} \rightarrow \mathbb{R}$ and closed set $$\Omega =\left\lbrace x \in\mathbb{R}^n \ |\ f(x) \leq 0 \right\rbrace$$ such that f is semi-analytic function and analytic inside $\mathbb{R}^{n} \setminus\Omega$ and $\text{Int }{\Omega}$. The Clarke's subdifferential set is the set given by $$ \partial f (x) = {\text{cl co}}\left\lbrace \lim_{k\in\mathbb{N}} \nabla f (x^{k}) \ |\ \text{whenever } x^{k} \rightarrow x^{*} \text{ and } \lim_{k\in\mathbb{N}} \nabla f (x^{k}) \text{ exist} \right\rbrace $$ for all $x \in \mathbb{R}^n.$ My question is about the context in which this multivalued function satisfies a weak notion of locally Lipschitz continuity for multifunctions; by that I mean that for all convex compact set $K$ there exist a $L>0$ such that for all $x\in K$ there exist a $\delta > 0 $ such that for all $y,z \in B(x,\delta) \cap K $ it holds $$ d(\partial f (y), \partial f (z)) = \inf_{g\in \partial f (y)} \inf_{g' \in \partial f (z) } \|g - g'\| < L \|y-z\|. $$ Note that when the function is locally Fréchet differentiable on an open set containing $\Omega$ with continuous second derivative, this is true. It seems to be true for all functions sufficiently regular. It might be true for piecewise smooth functions with sufficiently regular set of discontinuity of the derivative.
Is there a good context in which this is true?
