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
$$Z(S) =\{f \in C[0,\ 1] \mid f(x) = 0,\ \forall x \in S \}.$$
 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. 

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I can not understand how to proceed in this problem.Can I get some help.Thanks.