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 $f(x)=\lfloor x\rfloor$ or $f(x)=\Vert x\Vert .$ Consider the following examples. It can be noted that if $x$ is an integer, the function returns the same value. 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.

$x$	$f(x)=\lfloor x\rfloor$
$\text{-}1$	$\text{-}1$
$\text{-}0.75$	$\text{-}1$
$\text{-}0.5$	$\text{-}1$
$\text{-}0.25$	$\text{-}1$
$0$	$0$
$0.5$	$0$
$1$	$1$
$1.25$	$1$
$1.75$	$1$
$1.99$	$1$
$2$	$2$

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.