  Let $S$ be a closed subset of a Banach space $X$. Assuming that $S$ is
epi-Lipschitzian at $\bar{x}$ in the boundary $ \bd S$ of $S$, we show that $S$
is strictly Hadamard differentiable at $\bar{x}$ IFF the Clarke tangent cone
$T(S, \bar{x})$ to $S$ at $\bar{x}$ contains a closed hyperplane IFF the Clarke
tangent cone $T(\bd S, \bar{x})$ to $\bd S$ at $\bar{x}$ is a closed
hyperplane. Moreover when $X$ is of finite dimension, $Y$ is a Banach space and
$g: X \mapsto Y$ is a locally Lipschitz mapping around $\bar{x}$, we show that
$g$ is strictly Hadamard differentiable at $\bar{x}$ IFF $T(\mathrm{graph}\,g,
(\bar{x}, g(\bar{x})))$ is isomorphic to $X$ IFF the set-valued mapping
$x\rightrightarrows K(\gh g, (x, g(x)))$ is continuous at $\bar{x}$ and $K(\gh
g, (\bar{x}, g(\bar{x})))$ is isomorphic to $X$, where $K(A, a)$ denotes the
contingent cone to a set $A$ at $a \in A$.
