ISL 2019 A7

Let $\mathbb{Z}$ be the set of integers. We consider functions $f: \mathbb{Z} \rightarrow \mathbb{Z}$ satisfying $$f(f(x+y)+y)=f(f(x)+y)$$ for all integers $x$ and $y$. For such a function, we say that an integer $v$ is $f$-rare if the set $$X_{v}=\{x \in \mathbb{Z}: f(x)=v\}$$ is finite and nonempty. (a) Prove that there exists such a function $f$ for which there is an $f$-rare integer. (b) Prove that no such function $f$ can have more than one $f$-rare integer.