Clarke subgradients of stratifiable functions

We establish the following result: If the graph of a lower semicontinuous real extended valued function f : R-n -> R boolean OR{+infinity} admits a Whitney stratification ( so in particular if f is a semialgebraic function), then the norm of the gradient of f at x epsilon dom 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 - Lojasiewicz nequality for functions definable in an arbitrary o- minimal structure. It is worthwhile pointing outthat, 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.