$I\subseteq C_0(X)$ closed Ideal. Does for all $x\in X$ exist $f\in I$ such that $f(x)\neq 0$?

Question

$I$ a closed ideal in the Banach algebra $C_0(X)$, $X$ locally compact Hausdorff space. Is the claim correct: For all $x\in X$ exists $f\in I$ such that $f(x)\neq 0$?

I need this for a proof. But I have no idea if it's correct or false.