Graph of y = floor(x)

The floor function y = floor(x), also called the greatest integer function, rounds every real number down to the nearest integer less than or equal to x. Its graph looks like a staircase: flat horizontal segments at each integer level with jumps at every integer value of x.

The floor function is a step function with jump discontinuities at every integer. At each integer n, the function jumps from n-1 to n. This makes it a fundamental example of a discontinuous function and an important tool in number theory, computer science (integer division, array indexing), and digital signal processing (quantization).

Key properties: domain is all real numbers, range is all integers, floor(n) = n for any integer n, floor(x) <= x < floor(x) + 1. The function is constant on each interval [n, n+1) and is right-continuous. It has no derivative at integer values and a derivative of 0 everywhere else.