Floor function - Knowlpedia

For a real number $x$ , its floor $\lfloor x\rfloor$ is the unique integer satisfying

\lfloor x\rfloor\le x<\lfloor x\rfloor+1.

Existence follows from the Archimedean property and the well-ordering of nonnegative integers; uniqueness follows because distinct integers differ by at least one.

Examples and properties

$\lfloor2.3\rfloor=2$ , whereas $\lfloor-2.3\rfloor=-3$ . For every integer $k$ , $\lfloor x+k\rfloor=\lfloor x\rfloor+k$ . The function is constant on each interval $[k,k+1)$ and jumps at the integers. Holding a selected floor index fixed while differentiating another variable is different from differentiating the floor as a function of that variable.