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.