Spectrum of a ring of continuous functions

Question

The set $A=\{f:[0,1] \rightarrow \mathbb{R} : f$ is continuous$\}$ is a ring with the standard function addition and multiplication. Which are the prime ideals in $A$?

The only thing I've managed to observe is that the units of this ring are the functions that never take the value $0$, so every element of a prime ideal $P$ takes the value $0$ at least once.

Answer 1

fix $x_0$ ,then ideal $I_0=\{f\in A: f(x_0)=0\}$ is prime. For that if $fg\in I_0$ Then either $f$ or $g$ must be zero at $x_0$.

Edit1:i dont know if these are all ...

Edit 2:https://mathoverflow.net/questions/35793/prime-ideals-in-c0-1