Clarke Subgradients of Stratifiable Functions

We establish the following result: If the graph of a lower semicontinuous real-extended-valued function f:R ⁿ∪\+∞\ admits a Whitney stratification (so in particular if f is a semialgebraic function), then the norm of the gradient of f at x\,f relative to the stratum containing x bounds from below all norms of Clarke subgradients of f at x. As a consequence, we obtain a Morse–Sard type of theorem as well as a nonsmooth extension of the Kurdyka–Łojasiewicz inequality for functions definable in an arbitrary o-minimal structure. It is worthwhile pointing out that, even in a smooth setting, this last result generalizes the one given in [K. Kurdyka, Ann. Inst. Fourier (Grenoble), 48 (1998), pp. 769–783] by removing the boundedness assumption on the domain of the function.