Continuity of $f(x)=\left\lfloor \frac12 x -1\right\rfloor$

Question

How can you define the domain of a floor function, i.e. in interval notation, for which it is continuous? For example, for the following function: $$f(x)=\left\lfloor \frac12 x -1\right\rfloor$$

Answer 1

Hint: look for intervals of the form $(2n+2,2n+4)$ for $n \in \mathbb{Z}$.