Chapter 2.3, Problem 56E

To determine

To find: The correct choice for the interval on which the function $f\left(x\right)=\frac{1}{\sqrt{x}}$ is not continuous.

Answer to Problem 56E

The correct choice is (B).

Explanation of Solution

Given information: The function is $f\left(x\right)=\frac{1}{\sqrt{x}}$.

Check the continuity of the function $f\left(x\right)=\frac{1}{\sqrt{x}}$ with the help of graph.

To graph a function $y=\frac{1}{\sqrt{x}}$ , 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 $y=\frac{1}{\sqrt{x}}$ after the symbol ${\text{Y}}_1$ . Enter the keystrokes $\boxed{1}\boxed{÷}\boxed{2\text{nd}}\boxed{x^2}\boxed{\text{X}}\boxed{)}$.

The display will show the equation,

  ${\text{Y}}_{\text{1}}=1/\sqrt{\text{X}}$

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 $y=\frac{1}{\sqrt{x}}$ is continuous at every point of $x>0$ . So, the function $f\left(x\right)=\frac{1}{\sqrt{x}}$ is not continuous in the interval $\left[0,\infty \right)$.

Therefore, the correct option is (B).