Exercise 53 Page 224

The output of the greatest integer function, $f(x)=[[x]],$ is the greatest integer less than or equal to the input value $x.$ To help us understand how this function works we will first look at a few examples. We can think of it as rounding down the input value to the nearest integer. This results in a function that is divided into multiple intervals, where the function is constant within an interval.

Interval	$f(x)=[[x]]$
$\text{-}4\le x<\text{-}3$	$\text{-}4$
$\text{-}3\le x<\text{-}2$	$\text{-}3$
$\text{-}2\le x<\text{-}1$	$\text{-}2$
$\text{-}1\le x<0$	$\text{-}1$
$0\le x<1$	$0$
$1\le x<2$	$1$
$2\le x<3$	$2$
$3\le x<4$	$3$

Let's graph the function for $\text{-}4\le x<4.$

Recall that a closed dot corresponds to an end value that is included and an open dot means that the end value is not included.

Is It a Piecewise Function?

A piecewise function is any function defined by two or more equations, each applying to a different part of the domain. To show that the greatest integer function is a piecewise function, we can write the equation as constant functions defined on intervals.

Is It a Step Function?

A step function, on the other hand, is a special type of piecewise function characterized by a series of horizontal line segments. Let's once more study the graph of the greatest integer function.

The graph is a series of horizontal line segments. Therefore, the function $f(x)=[[x]]$ is a step function.