INMO 2012 Problem 6

Let $f: \mathbb{Z} \rightarrow \mathbb{Z}$ be a function satisfying $f(0) \neq 0, f(1)=0$ and

(i) $f(x y)+f(x) f(y)=f(x)+f(y)$;

(ii) $(f(x-y)-f(0)) f(x) f(y)=0$,

for all $x, y \in \mathbb{Z}$, simultaneously.

(a) Find the set of all possible values of the function $f$.

(b) If $f(10) \neq 0$ and $f(2)=0$, find the set of all integers $n$ such that $f(n) \neq 0$.