Question 46
Determine all functions $f$, defined on the set $\mathbb{Z}$ of integers and with values in the set $\mathbb{R}$ of real numbers, that simultaneously satisfy the following properties:
- for every pair of integers $(m, n)$ with $m \lt n$ we have $f(m) \lt f(n)$;
- for every pair of integers $(m, n)$ there exists an integer $k$ such that $f(m) - f(n) = f(k)$.
- for every pair of integers $(m, n)$ with $m \lt n$ we have $f(m) \lt f(n)$;
- for every pair of integers $(m, n)$ there exists an integer $k$ such that $f(m) - f(n) = f(k)$.