Characteristics of Floor and Ceiling Function

Theorem
Let $f: \R \to \Z$ be an integer-valued function which satisfies both of the following:
 * $(1): \quad f \left({x + 1}\right) = f \left({x}\right) + 1$
 * $(2): \quad \forall n \in \Z_{> 0}: f \left({x}\right) = f \left({f \left({n x / n}\right)}\right)$

Then either:
 * $\forall x \in \Q: f \left({x}\right) = \left \lfloor{x}\right \rfloor$

or:
 * $\forall x \in \Q: f \left({x}\right) = \left \lceil{x}\right \rceil$