# Distance to integers

Define the function $\mathrm{dist}_{\mathbb{Z}} \colon \mathbb{R} \rightarrow \mathbb{R}$ by $$\mathrm{dist}_{\mathbb{Z}}(x)=\inf_{n \in \mathbb{Z}} |x-n|,$$ where $\inf$ denotes the infimum. This function can be computed using the floor and ceiling functions: $$\mathrm{dist}_{\mathbb{Z}}(x)=\min \left( x - \lfloor x \rfloor, \lceil x \rceil - x \right).$$