The greatest integer function, also known as the floor function, assigns the largest integer that is less than or equal to the value of x. This function is usually written as f(x)= ⌊ x ⌋ or f(x)= ||x||. Consider the following examples. ⌊ - 2 ⌋ & = - 2 ⌊ - 1.5 ⌋ & = - 2 ⌊ -0.01 ⌋ & = -1 ⌊ 1 ⌋ & = 1 ⌊ 1.25 ⌋ & = 1 It can be noted that if x is an integer, the function returns the same value. ⌊ - 2⌋ = - 2 ⌊ 0⌋ = 0 ⌊ 1 ⌋ = 1 Otherwise, it returns the closest integer at the left of x in a number line.

Considering more values can help understand how to draw the graph of the function.

x	f(x)=⌊ x ⌋
- 1	- 1
- 0.75	- 1
- 0.5	- 1
- 0.25	- 1
0	0
0.5	0
1	1
1.25	1
1.75	1
1.99	1
2	2

From the table above, it can be seen that the function only changes its value when a new integer is reached. It can be noted that the greatest integer function is a step function. Its graph is presented as follows.

The domain of the function is the set of all real numbers, but its range is the set of integers. A filled point indicates that the point is part of the graph, while an open point indicates that it is not.