It is clear that each maximal ideal in ring of continuous functions over $[0,1]\subset \mathbb R$ corresponds to a point and vice-versa.

So, for each ideal $I$ define $Z(I) =$ {$x\in [0,1]$ | $f(x)=0, \forall f \in I$}. But map $I\to Z(I)$ from ideals to closed sets isn't injection! (Consider ideal $J(x_0)=${$f$|$f(x)=0, \forall x\in$ some closed interval which contains $x_0$})

How can we describe ideals in C([0,1])? Is it true that prime ideal is maximal for this ring?