Gateaux and Hadamard differentiability via directional differentiability

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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 ∈ X except a σ-directionally porous set: If the one-sided Hadamard directional derivative f'ₕ₊(x,u) exists in all directions u from a set Sₓ ⊂ 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 Sₓ equals X and Y =. 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.