Can someone just confirm the following, I think I already managed to prove it but it seems so important that I want to know if I did some mistake...
If $f_i:U\subseteq E\rightarrow F$ is a sequence of continuously differentiable functions between (open subsets of) Banach spaces and $f_i$ as well as $D(f_i)$ converge uniformly on $U$, then the limit $f$ is also continuously differentiable with $D(f)(x)=\lim_i D(f_i)(x)$.
BTW, I proved it via the Mean Value Theorem obtained by an application of Hahn Banach (I found this here if anyone is interested).