Showing a function is not continuous in the one-norm 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.

If I understand your question and you mean $L^1$ norm (you also mention sup norm and $F$ is continuous with respect to that, which is perhaps point of problem), you could let $f_n$ be $0$ on $[0,1/2-1/n^2]$, the line from $0$ to $n$ on $[1/2-1/n^2,1/2]$, the line from $n$ to $0$ on $[1/2,1/2+1/n^2]$, and $0$ on $[1/2+1/n^2]$.  Then $f_n \rightarrow 0$ in $L^1$, but $F(f_n)=n$.
Basically any function that is zero except for a "high spike of small area" will work.
