Gâteaux and Hadamard Differentiability via Directional Differentiability

Let

be a separable Banach space,

Y Y

a Banach space and

f: X \to Y f : X → Y

an arbitrary mapping. Then the following implication holds at each point

x\in X x ∈ X

except a

\sigma σ

-directionally porous set: If the one-sided Hadamard directional derivative

f'_{H+}(x,u) f H + ′ ( x , u )

exists in all directions

u u

from a set

S_x \subset X S x ⊂ X

whose linear span is dense in

X X

, then

f f

is Hadamard differentiable at

x x

. This theorem improves and generalizes a recent result of A. D. Ioffe, in which the linear span of

S_x S x

equals

X X

and

Y = \mathbb{R} Y = R

. An analogous theorem, in which

f f

is pointwise Lipschitz, and which deals with the usual one-sided derivatives and Gâteaux differentiability is also proved. It generalizes a result of D. Preiss and the author, in which

f f

is supposed to be Lipschitz.