Chapter 1, Problem 7RWDT

To determine

To Give: The example of a composite function $g\circ f$ that is continuous at $x=0$ ,

where $f$ is not continuous at $0$ .

Answer to Problem 7RWDT

The function $\left(g°f\right)\left(x\right)=x^2$ where $f\left(x\right)=\frac{1}{x}$ , $g\left(x\right)=\frac{1}{x^2}$ is a example of a composite function $g\circ f$ that is continuous at $x=0$ , where $f$ is not continuous at $0$ .

Explanation of Solution

Given:

Give the example of a composite function $g\circ f$ that is continuous at $x=0$ ,

where $f$ is not continuous at $0$ .

Let $f\left(x\right)=\frac{1}{x}$ , $g\left(x\right)=\frac{1}{x^2}$

  $f\left(x\right)$ is not continuous at $x=0$ (not defined at $x=0$ )

Find the value of $\left(g°f\right)\left(x\right)=g\left(f\left(x\right)\right)$ :

  $\begin{array}{c}g\left(\frac{1}{x}\right)=\frac{1}{{\left(\frac{1}{x}\right)}^2}\\ =x^2\end{array}$

Thus, $\left(g°f\right)\left(x\right)=x^2$ .

Here, $\left(g°f\right)\left(x\right)=x^2$ is continuous at $x=0$