Characteristics of Floor and Ceiling Function/Real Domain

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 it is not necessarily the case that either:
 * $\forall x \in \R: f \left({x}\right) = \left \lfloor{x}\right \rfloor$

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

Proof
Let $h: \R \to \R$ be a real function such that for all $x, y \in \R$:

Consider the integer-valued function $f: \R\to \Z$ defined as:
 * $f \left({x}\right) = \left \lfloor{h \left({x}\right)}\right \rfloor$

Then $f$ satisfies $(1)$ and $(2)$.

But it may be that $h \left({x}\right)$ is "unbounded and highly erratic" when $0 < x < 1$.