Prove the supremum norm on the space $C[0,1]$ is not differentiable at any element $x$ for which there are two point $t$ in $[0,1]$ where $|x(t)|=\|x\|$.
My attemp: Suppose that $f:C[0,1]\to\mathbb{R}$, is differentiable for any point $[0,1]$. If we have two point $t_1,t_2\in[0,1]$ where $|x(t)|=\|x\|$, then the function $F(t)=f(v+tu)$ will be differentiable on $t=0$ for any element $v\in C[0,1]$.But know I don't see how take a contradiction for the function $f$, and therefore show that is not differentiable.