Is the sup norm a differentiable function on $\mathcal{C}[a,b]$?
Hint: Consider $F:[a,b] \rightarrow \mathbb{R}$ defined by $F(u) = max_{t \in [a,b]} \lvert u(t) \rvert$. Can you find a function $h \in \mathcal{C}[a,b]$ such that $\lvert\lvert u + h \rvert\rvert \sim \lvert\lvert u-h \rvert\rvert \sim \lvert\lvert u \rvert\rvert + \varepsilon?$ If so, what does this imply?
I guess it is clear that $h := u + \varepsilon$ fullfilles these criteria, but I do not see how this should help here.