Let $f : \mathbb { Z } \rightarrow \mathbb { Z }$ be a function satisfying $f ( 0 ) \neq 0 = f ( 1 )$. Assume also that $f$ satisfies equations $( \mathbf { A } )$ and $( \mathbf { B } )$ below. $$\begin{aligned}
f ( x y ) & = f ( x ) + f ( y ) - f ( x ) f ( y ) \\
f ( x - y ) f ( x ) f ( y ) & = f ( 0 ) f ( x ) f ( y )
\end{aligned}$$ for all integers $x , y$. (i) Determine explicitly the set $\{ f ( a ) : a \in \mathbb { Z } \}$. (ii) Assuming that there is a non-zero integer $a$ such that $f ( a ) \neq 0$, prove that the set $\{ b : f ( b ) \neq 0 \}$ is infinite.
Let $f : \mathbb { Z } \rightarrow \mathbb { Z }$ be a function satisfying $f ( 0 ) \neq 0 = f ( 1 )$. Assume also that $f$ satisfies equations $( \mathbf { A } )$ and $( \mathbf { B } )$ below.
$$\begin{aligned}
f ( x y ) & = f ( x ) + f ( y ) - f ( x ) f ( y ) \\
f ( x - y ) f ( x ) f ( y ) & = f ( 0 ) f ( x ) f ( y )
\end{aligned}$$
for all integers $x , y$.\\
(i) Determine explicitly the set $\{ f ( a ) : a \in \mathbb { Z } \}$.\\
(ii) Assuming that there is a non-zero integer $a$ such that $f ( a ) \neq 0$, prove that the set $\{ b : f ( b ) \neq 0 \}$ is infinite.