Set of functions open or/and closed

Question

Say we have the set $A=\{f \in C[0,1]: f(0)=0\}$. I need to show if this set is open or and closed with respect to the supnorm. I am only allowed to show this with the open/closed balls definition and can’t use other tools. So I tried this but it did not got me far.

Let f be arbitrary. Choose $r=$... Then for $g \in C[0 ,1]$ we have:

$\sup_{t \in [0,1]} |g-f| \leq r.$ I already get stuck here. I do not have an idea how I can use f(0)=0 such that this will lead me to $g(0)=0$ by choosing a handy $r$.

Answer 1

Hint: Show that complement is open.

Spoiler (hover over the block below):

Let $g$ be in the complement. Then, we know that $g(0)\neq 0.$ Take $r=|g(0)|/2.$ Then, if $h\in B(r,g),$ we can see that $$|h(0)|=|h(0)+g(0)-g(0)|\geq |g(0)|-|h(0)-g(0)|\geq |g(0)|/2>0.$$ So, $B(r,g)$ is contained in the complement. This means that the complement is open, so the original set is closed.

I know you aren't allowed to use limit points, but if you're familiar with them, it's very easy to see right away that the set must be closed. This, at least, tells you what you want to prove.