# Analytically characterizing the set of integers at which the floor and ceiling of two affine functions with the same slope are equal.

Assume that $$a>0$$, $$b_0>0$$, $$b_1>0$$ and that $$-2. Define $$f_0(x) = \lfloor b_0-ax \rfloor$$ and $$f_1(x) = \lceil b_1-ax \rceil$$. Finally, define the following set:

$$\mathcal{D} = \Big\{n\in\{0,1,2,...\}: f_0(n)=f_1(n) \Big\}$$

Is it possible to analytically characterize $$\mathcal{D}$$?

1. If it is significantly easier to analytically characterize $$\Big\{n\in\{0,1,2,...\}: f_0(n)f_1(n) \Big\}$$ please feel free to replace $$\mathcal{D}$$ with one of the above sets. (I would appreciate your answer just as much! Which is a lot; thank you so much for taking the time to even read this.)