 Hadamard differentiability

A map on a subset of a normed vector space is Hadamard differentiable at $s$ if its directional difference quotients converge to a continuous linear map uniformly along all convergent sequences of directions: $t_j\to 0$, $a_j\to a$, and $s+t_j{a}_j$ in the domain imply $(\Phi (s+t_j{a}_j)-\Phi (s))/t_j\to {\dot{\Phi }}_s(a)$. Tangential differentiability restricts the limit directions to a specified subspace. This is the regularity used by the functional delta method.