I would like to find a proof on the following proposition.
Let $X$ and $Y$ be two Banach spaces and $O$ an open set of $X$ and $\overset{\sim}{X}$ be a dense subspace e of $X$. Let $F : O \to Y$ be a continuous application. We suppose that $ \forall x \in \overset{\sim}{X} \cap O$ there exists a linear application $L : X\to Y$ such that :
$$\forall v \in \overset{\sim}{X},\; \lim_{t\to 0} \frac{F(x+tv)-F(x)}{t}=L(x)v $$
If the application $x \to L(x)$ is continuous on $\overset{\sim}{X} \cap O$ then $F \in C^1(O)$ and $DF(x) = L(x)$.