Nearest integer - Knowlpedia

A nearest integer to $x\in\mathbb R$ is an integer $n$ minimizing the absolute value $|x-n|$ . It always satisfies $|x-n|\le1/2$ . The minimizer is unique unless $x$ is halfway between two consecutive integers.

A convention

One deterministic rule is $n(x)=\lfloor x+1/2\rfloor$ , which resolves a tie toward the larger integer. Other rules, such as rounding ties to an even integer, must be specified if used. A nearest nonzero integer minimizes over $\mathbb Z\setminus\{0\}$ and can differ from ordinary rounding near zero; a tie between $-1$ and $1$ occurs at zero.

Example

The nearest integer to $2.3$ is $2$ , while both $2$ and $3$ minimize distance to $2.5$ . Rounding is a discrete selection operation, so differentiability cannot be assumed at the selection boundaries.