Question 175
Let $\mathbb{Z}$ be the set of integers, and let $f: \mathbb{Z} \to \mathbb{Z}$ be a function. Prove that there are infinitely many integers $c$ such that the function $g: \mathbb{Z} \to \mathbb{Z}$ defined by $g(x) = f(x) + c x$ is not bijective.
*Note*: A function $g: \mathbb{Z} \to \mathbb{Z}$ is *bijective* if for every integer $b$, there exists exactly one integer $a$ such that $g(a) = b$.
*Note*: A function $g: \mathbb{Z} \to \mathbb{Z}$ is *bijective* if for every integer $b$, there exists exactly one integer $a$ such that $g(a) = b$.