General topology – Show that $ Z (I_ {A}) = A $

Problem:

Consider $ X $ a compact Hausdorff room with $ C (X) $ the set of continuous functions $ X $, If $ A $ is closed subset of $ X $, define $ I_ {A} = {f in C (X) | f | _ {A} = 0 } $,
and $ Z (I): = {x in X | f (x) = 0 $ for all $ f in I } $,

$ 1 $ ) Show that $ Z (I_ {A}) = A $,

First if $ x in A $then for everyone $ f in I_ {A}, f (x) = 0 $, so $ x in Z (I_ {A}) $, The other direction confuses me (it could be wrong). Is not it possible to have an ideal of function? $ C (X) $ Be zero $ A $ but also on other sets? Note appreciated. I know that $ I_ {A} $ is an ideal that is closed in relation to the norm topology (the sup norm).