Definition:Distance to Nearest Integer Function

Definition
The nearest integer function $\left\Vert{\cdot}\right\Vert : \R \to \left[{0 \,.\,.\, \dfrac 1 2}\right]$ is defined in the following ways:

Equivalence of Definitions
See Equivalence of Definitions of Nearest Integer Function for a proof that these definitions are equivalent.

Also denoted as
The notation $\left\Vert{\cdot}\right\Vert_{\R / \Z}$ is also in use.