I have the following question:
Consider the normed space $(C([0,1]),||\cdot||_{\infty})$. Let $x_{0}\in [0,1]$ and define $F:C([0,1])\to \mathbb{R}$ by
$F(f)=f(x_{0})$
Show that $F$ is discontinuous with respect to $||\cdot||_{1}$.
I realise that I simply need to find a counterexample where $f_{n}\to f$ with respect to the one-norm, but where $F(f_{n})\not \to F(f)$. However, I see no way in which to do this. I hate being lazy with this so would appreciate a small push in the right direction.