Let F F be a continuous mapping from an open subset D D of a separable Banach space X X into a Banach space Y Y . We show that if the one sided directional derivative D x + F ( a ) D_x^ + F(a) of F F at a a in the direction x x exists for each ( a , x ) (a,x) from a dense G δ {G_\delta } subset S S of an open set D × U ⊂ X × X D \times U \subset X \times X , then F F is Gâteaux differentiable on a dense G δ {G_\delta } subset of D D . Similar results are obtained for Fréchet differentiability when X X is finite-dimensional and for w ∗ {w^ * } -Gâteaux differentiability.