INMO 2023 Problem 3

Let $\mathbb{N}$ denote the set of all positive integers. Find all real numbers $c$ for which there exists a function $f: \mathbb{N} \rightarrow \mathbb{N}$ satisfying:

(a) for any $x, a \in \mathbb{N}$, the quantity $\frac{f(x+a)-f(x)}{a}$ is an integer if and only if $a=1$;

(b) for all $x \in \mathbb{N}$, we have $|f(x)-c x|<2023$.