Abstract
Let X be a separable Banach space, Y a Banach space and f : X -> Y an arbitrary mapping. Then the following implication holds at each point x is an element of X except a sigma-directionally porous set: If the one-sided Hadamard directional derivative f(H+)'(x,u) exists in all directions u from a set S-x subset of X whose linear span is dense in X, then f is Hadamard differentiable at x.
This theorem improves and generalizes a recent result of A. D.
Ioffe, in which the linear span of S-x equals X and Y = R. An analogous theorem, in which f is pointwise Lipschitz, and which deals with the usual one-sided derivatives and Gateaux differentiability is also proved.
It generalizes a result of D. Preiss and the author, in which f is supposed to be Lipschitz.