Differentiability via one-sided directional derivatives

Abstract:

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