> 
> Let $E = \mathcal{C}([0,1],\mathbb{R})$ a Banach space of continuous fonctions mapping from $[0,1]$ to \mathbb{R}, with the norm $||f|| = \underset{t \in [0,1]}{\sup}|f(t)|$. Let $\Psi : E \rightarrow E$ defined as $\Psi(f) = \sin(f(t)), \forall f \in E, \forall t \in [0,1] $.  
> 
>  1) Verify that $\Psi$ is continuous  
> 
>  2) Show that $\Psi$ is Frechet differentiable on $E$, and find its differential.
> 
> 
> 

I trying to solve this exercise, but I struggle a lot. I think that to show its continuity, you have to show that $\Psi$ is a composite function of two continuous functions. Once you manage to describe $\Psi$ with the composition of two functions, answering the second question might be more easier.

The issue I get with the composition, is that my intuition first tells me that $$\Psi(f) = g \circ u $$ where $u:E \rightarrow \mathbb{R}$ such that $u(f) = f(t)$ and $g: \mathbb{R} \rightarrow \mathbb{R}$ such that $g(x)=sin(x)$ Now this is obviously false as $\Psi$ maps to $E$ and not $\mathbb{R}$.

Can someone help me to find the two functions that fit?