Show that evaluation is continuous on $C[0,1]$

Question

Choose and fix a point $c \in [0,1]$, and consider the function $E: C[0,1] \rightarrow R $ given by $E(f) = f(c)$. Show that $E$ is continuous.

My question is what does fix a point mean and what is $E(f)$.

Answer 1

Note that if $\|f-g\| < \epsilon$ then $|f(c)-g(c)| < \epsilon$.

To elaborate slightly: $|E(f)-E(g)| \le \| f-g \|$, hence $E$ is Lipschitz continuous.

Answer 2

It is not clear which topology you have on $C[0,1]$.

1. if we have the maximum-norm $\|*\|_{\infty}$ on $C[0,1]$,then $E$ is continuous.

2. if we have the $L^1$-norm $\|*\|_{1}$ on $C[0,1]$,then $E$ is not continuous.