Question 30
Let $\mathbb{N}$ denote the set of all positive integers. Find all real numbers $c$ for which there exists a function $f: \mathbb{N} \to \mathbb{N}$ 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 $|f(x) - cx| \lt 2023$.
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 $|f(x) - cx| \lt 2023$.