We establish the following result: if the graph of a (nonsmooth) real-extended-valued function $f:mathbb{R}^{n}to mathbb{R}cup{+infty}$ is closed and admits a Whitney stratification, then the norm of the gradient of $f$ at $xin{dom}f$ 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-L ojasiewicz inequality for functions definable in an arbitrary o-minimal structure.
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