Question 58
Let $f: \mathbb{Z} \rightarrow \mathbb{Z}$ be a function satisfying $f(0) \neq 0$, $f(1)=0$ and
(i) $f(xy)+f(x)f(y)=f(x)+f(y)$;
(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 $f(n) \neq 0$.
(i) $f(xy)+f(x)f(y)=f(x)+f(y)$;
(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 $f(n) \neq 0$.