We establish the following result: if the graph of a (nonsmooth) real-extended-valued function f : R n → R ∪ {+∞} is closed and admits a Whitney stratification, then the norm of the gradient of f at x ∈ 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 some Morse-Sard type theorems as well as a nonsmooth Kurdyka-Lojasiewicz inequality for functions definable in an arbitrary o-minimal structure.

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PREFACEThe purpose of these Lecture Notes is to construct a theory of real manifolds equipped with "algebraic" structures. Real nonsingular algebraic varieties display apparently singular phenomena, and this suggests we should not expect a unified theory to emerge from them.As we shall see, however, the objects we consider in these Notes form a broader class than the real nonsingularalgebraic varieties.A Nash manifolds and C r Nash maps that we are interested in.The compact case was already studied well. In [N] Nash showed that a compact C l manifold M can be imbedded in a Euclidean space m n so that the image is a C W Nash submanifold of ]Rn. He proved also that such a C W Nash manifold structure on M is unique up to C W Nash diffeomorphism. Hence we can endow a compact C l manifold with "algebraic"properties, which appears to contribute to differential topology.

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