USAJMO 2025 Problem 1

Let $\mathbb Z$ be the set of integers, and let $f\colon \mathbb Z \to \mathbb Z$ be a function. Prove that there are infinitely many integers $c$ such that the function $g\colon \mathbb Z \to \mathbb Z$ defined by $g(x) = f(x) + cx$ is not bijective. Note: A function $g\colon \mathbb Z \to \mathbb Z$ is bijective if for every integer $b$ , there exists exactly one integer $a$ such that $g(a) = b$ .