Chapter 2.3, Problem 7E

To determine

To find:The points of continuity and discontinuity of the function $y=\frac{\left|x\right|}{x}$ . Also, write the type of each discontinuity.

Answer to Problem 7E

The function $y=\frac{\left|x\right|}{x}$ is continuous for all the values of $x$ except $x=0$ with jump discontinuity.

Explanation of Solution

Given information:The functionis $y=\frac{\left|x\right|}{x}$ .

Calculation:

To graph a function $y=\frac{\left|x\right|}{x}$ , follow the steps using graphing calculator.

First press “ON” button on graphical calculator, press $\boxed{\text{Y}=}$ key and enter right hand side of the equation $y=\frac{\left|x\right|}{x}$ after the symbol ${\text{Y}}_1$ . Enter the keystrokes $\boxed{(}\boxed{\text{MATH}}\boxed{\to }\boxed{1}\boxed{\text{X}}\boxed{)}\boxed{)}\boxed{÷}\boxed{\text{X}}$ .

The display will show the equation,

  ${\text{Y}}_{\text{1}}=\left(\text{abs}\left(\text{X}\right)\right)/\text{X}$

Now, press the $\boxed{\text{GRAPH}}$ key and $\boxed{\text{TRACE}}$ key to produce the graph of given function in standard window as shown below.

  

Figure(1)

As observed from graph, the function $y=\frac{\left|x\right|}{x}$ is continuous at every point of $x$ except at $x=0$ .

The function $y=\frac{\left|x\right|}{x}$ is discontinuous at $x=0$ with jump discontinuity as the left hand limit of function is $-1$ and right hand limit of function is $1$ at $x=0$ .

Therefore, the function $y=\frac{\left|x\right|}{x}$ is continuous for all the values of $x$ except $x=0$ with jump discontinuity.