I came across the following exercise, namely proofing that for the minimumfunction f(a,b) = min(a,b) for $ x = (a,b)^T \in \mathbb{R}^2$, it holds for every point that
sup $|f(x+s)-f(x) - Ms| = O(||s||^2)$ for $||s|| \to 0$ where $M$ is an element of Clarke's generalized differential at x+s.
It is clear that at points where $x_1 \neq x_2$, this holds (as then f is linear, the only element in Clarke's differential is its Jacobian and everything cancels. However, I seem to miss the idea for $x_1 = x_2$.
This generalizes to arbitrary piecewise linear functions, hence the title.
Any help is appreciated!
I've found a good source online to answer this question: It holds in general for $PC^2$ (piecewise $C^2$-functions) that they are first order semismooth. A proof is given in "Semismooth Newton Methods for Variational Inequalities and Constrained Optimization Problems in Function Spaces", (M. Ulbrich) Proposition 2.2.6.
The basic idea (that was missing for me) was to write $M$ as a convex combination of the elements in the B-differential and then to use Taylor to get an upper bound.