  Using a geometric argument, we show that under a reasonable continuity
condition, the Clarke subdifferential of a semi-algebraic (or more generally
stratifiable) directionally Lipschitzian function admits a simple form: the
normal cone to the domain and limits of gradients generate the entire Clarke
subdifferential. The characterization formula we obtain unifies various
apparently disparate results that have appeared in the literature. Our
techniques also yield a simplified proof that closed semialgebraic functions on
$\R^n$ have a limiting subdifferential graph of uniform local dimension $n$.
