So the function given is $$ g{_r}: \mathbb{Z}\rightarrow\mathbb{Z}, \: x \mapsto \big\lfloor\dfrac{x}{r}\big\rfloor, \quad \quad \text{while} \: r\in\mathbb{N}.$$
The original question was to prove or disprove the injectivity or surjectivity of this function. I have already proven that for $r=1$, this function is injective and for $r>1$ it is not injective.
Now I have to check if the function is surjective for $r>1$.
My thought is that we assume that the function is surjective, then we have to show that for every $\left\lfloor\dfrac{x}{r}\right\rfloor\in\mathbb{Z}$ exists an $x \in\mathbb{Z}$. How can I prove (or disprove) this? Are there some transformations that I can do to the floor function?
Hint: $\lfloor\frac{rt}{r}\rfloor=\lfloor t\rfloor$