I have the following problem:
Suppose $f\in C^1(\mathbb{R},\mathbb{R})$. Let $C([0, 1])$ be the space of continuous functions with norm $||u||_{\infty} = \max_{x \in [0, 1]} |u(x)|$. Show that the map $F: C([0, 1]) \rightarrow C([0, 1])$ defined by $F(u)(x) = f(u(x))$ is $C^1$ and compute its derivative.
It seems pretty simple to show that $F$ is continuous and since $f$ is $C^1$, $$ \lim_{h \rightarrow 0} \frac{F(u+h)(x) - F(u)(x)}{h} = \lim_{h \rightarrow 0} \frac{f(u(x) + h) - f(u(x))}{h} $$ tells us that $F$ is $C^1$ also. My questions are:
1) When you compute the derivative by the chain rule you get $\dot F(u)(x) = \dot f(u(x)) \dot u(x)$, but since $C([0, 1])$ is only a space of continuous functions, it's not necessarily true that $u(x)$ will be differentiable. So is my formula wrong?
2) Nowhere did I use the norm given for the problem. This combined with 1) leads me to believe I am missing some major parts of the proof and there is a lot more subtlety involved. Some help would be appreciated.