Chapter 1.3, Problem 108E

To determine

To explain:

Difference of graph of greatest integer and the graph of a line with a slope of zero.

Answer to Problem 108E

The graph of greatest integer function differs from the graph of a line with a slope of zero, which is line parallel to the $x-$ axis.

Explanation of Solution

The greatest integer function can be represented by using a piecewise-defined function is explained with the help of example,

Consider the function, $f\left(x\right)=〚x〛$

  $f\left(0\right)=0,f\left(x\right)=0$ for $0<x<1$

  $f\left(1\right)=1,f\left(x\right)=1$ for $1<x<2$

  $f\left(2\right)=2,f\left(x\right)=2$ for $2<x<3$

So this can be represented as a piecewise defined function as below,

  $f\left(x\right)=\{\begin{array}{l}⋮\\ 0for0<x<1\\ 1for1<x<2\\ 2for2<x<3\\ ⋮\end{array}$

So, Graph of thisfunction is below,

  

Thus, the graph of greatest integer function is a combination of horizontal line segments of slope zero.

So, it differs from the graph of a line with a slope of zero, which is line parallel to the $x-$ axis.