Chapter 2.7, Problem 52E

Where is the greatest integer function f(x) = [[ x ]] not differentiable? Find a formula for[' and sketch its graph.

To determine

To find: The points where the greatest integer function $f\left(x\right)=〚x〛$ is not

differentiable and then to find the formula for $f^{\prime }$ and sketch the graph.

Answer to Problem 52E

The greatest integer function is not differentiable at $x\in \mathbb{Z}$

The formula for the derivative function is $f^{\prime }\left(x\right)=\{\begin{array}{cc}0& \text{if }x\cancel{\in }\mathbb{Z}\\ \text{undefined}& \text{if }x\in \mathbb{Z}\end{array}$.

Explanation of Solution

Result Used:

The greatest integer function is defined as follows,

$f\left(x\right)=\{\begin{array}{l}y\text{ if }x\cancel{\in }\mathbb{Z}:y\text{ is nearest interger less than }x\\ x\text{ if }x\in \mathbb{Z}\end{array}$

Calculation:

Obtain the points where the greatest integer function $f\left(x\right)=〚x〛$ is not differentiable.

The derivative of the greatest integer function is calculated as follows,

Case 1: x is not an integer.

$\begin{array}{c}{f}^{\prime }\left(x\right)=\underset{h\to 0}{\lim }\frac{f\left(x+h\right)-f\left(x\right)}{h}\\ =\underset{h\to 0}{\lim }\frac{〚x+h〛-〚x〛}{h}\end{array}$

Since x is not an integer, $〚x+h〛=〚x〛$.

$\begin{array}{c}{f}^{\prime }\left(x\right)=\underset{h\to 0}{\lim }\frac{〚x〛-〚x〛}{h}\\ =\underset{h\to 0}{\lim }\frac{0}{h}\\ =0\end{array}$

Thus, $f^{\prime }\left(x\right)=0$

Case 2: x is an integer.

The left hand derivative of the function is computed as follows,

$\begin{array}{c}{f^{\prime }}_{-}\left(x\right)=\underset{h\to 0^{-}}{\lim }\frac{f\left(x+h\right)-f\left(x\right)}{h}\\ =\underset{h\to 0^{-}}{\lim }\frac{〚x+h〛-〚x〛}{h}\end{array}$

Since h approaches 0 from left, $x+h<x$.

This implies that, $〚x+h〛=x-1$.

$\begin{array}{c}\underset{h\to 0^{-}}{\lim }\frac{〚x+h〛-〚x〛}{h}=\underset{h\to 0^{-}}{\lim }\frac{x-1-x}{h}\\ =\frac{-1}{\underset{h\to 0^{-}}{\lim }h}\\ =\frac{-1}{0}\end{array}$

Thus, the left hand derivative of the function does not exist. This follows that, $f^{\prime }\left(x\right)$ does not exist.

Therefore, the greatest integer function is not differentiable at the integer points.

By case (1) and (2), it is defined that the formula for the derivative is $f^{\prime }\left(x\right)=\{\begin{array}{cc}0& \text{if }x\cancel{\in }\mathbb{Z}\\ \text{undefined}& \text{if }x\in \mathbb{Z}\end{array}$.

Graph:

Use the above information and trace the graph of $f^{\prime }$ as shown below in Figure 1,

From the Figure 1, it is clear that the greatest integer function is discontinuous at integer points.