Signum and greatest integer functions

The signum function f : R → R gives the sign of x: f(x) = 1 if x > 0, f(0) = 0, and f(x) = −1 if x < 0.

The domain of the signum function is R and its range is {−1, 0, 1}. Its graph is two horizontal rays at heights 1 and −1, with open ends at x = 0, and the single point (0, 0).

For x ≠ 0 the signum function equals |x|/x, which is 1 for positive x and −1 for negative x; the formula |x|/x itself is undefined at 0.

The greatest integer function f : R → R, f(x) = [x], gives the greatest integer less than or equal to x. [3.7] = 3, [5] = 5 and [−2.4] = −3, not −2.

For every integer n, [x] = n exactly when n ≤ x < n + 1. So [x] = 0 on [0, 1), [x] = 1 on [1, 2) and [x] = −1 on [−1, 0). Its domain is R and its range is Z.

The graph of [x] is a staircase of horizontal steps of length 1. Each step includes its left end (filled dot) and excludes its right end (hollow dot), where the value jumps up by 1.

For every real x, [x] ≤ x < [x] + 1, and [x] = x exactly when x is an integer.