Let $C[0,1]$ be the ring of continuous real-valued functions on $[0,1]$, with addition and multiplication defined pointwise. For any subset $S$ of $C[0,1]$ let $Z(S)=\{f\in C[0,1]: f(x)=0 \text{ for all }x\in S\}$. Then which of the following statements are true?
(a) If $Z(S)$ is an ideal in $C[0,1]$ then $S$ is closed in $[0,1]$.
(b) If $Z(S)$ is a maximal ideal then $S$ has only one point.
(c) If $S$ has only one point then $Z(S)$ is a maximal ideal.
(a) Not necessary: if I take $S=(1/2,1/3)$ still $Z(S)$ is an ideal.
(b) I know that maximal ideals in $C[0,1]$ come in this way (I don't know the proof rigorously), i.e. $C_a=\{f\in C[0,1]:f(a)=0\}$ so $S$ may be finite or countable set? So I guess $(b)$ is a true statement and for the same reason $(c)$ is also true. But I will be happy if someone can explain me a bit about (b) and (c). Thank you.