Školjka

%V0 Let $f$ be a non-constant function from the set of positive integers into the set of positive integer, such that $a-b$ divides $f\!\left(a\right)-f\!\left(b\right)$ for all distinct positive integers $a$, $b$. Prove that there exist infinitely many primes $p$ such that $p$ divides $f\!\left(c\right)$ for some positive integer $c$. Proposed by Juhan Aru, Estonia