Let S be a closed subset of a Banach space X. Assuming that S is epi-Lipschitzian atx in the boundary bdry S of S, we show that S is strictly Hadamard differentiable atx IFF the Clarke tangent cone T (S,x) to S atx contains a closed hyperplane IFF the Clarke tangent cone T (bdry S,x) to bdry S atx is a closed hyperplane. Moreover when X is of finite dimension, Y is a Banach space and g : X → Y is a locally Lipschitz mapping aroundx, we show that g is strictly Hadamard differentiable atx IFF T (graph g, (x, g(x))) is isomorphic to X IFF the set-valued mapping x ⇒ K(graph g, (x, g(x))) is continuous atx and K(graph g, (x, g(x))) is isomorphic to X, where K(A, a) denotes the contingent cone to a set A at a ∈ A.