Chapter 1.3, Problem 63E

To determine

To show : function $f$ continuous on and interval, then so $\left|f\right|$ continuous.

Explanation of Solution

Given information :

function $f$ continuous.

Assume a function $g\left(x\right)=\left|x\right|$ which is continuous over all real numbers.

Since, $f$ is given to be continuous.

Replace $x$ by $f\left(x\right)$ in function $g\left(x\right)=\left|x\right|$ .

  $\begin{array}{c}g\left(f\left(x\right)\right)=\left|f\left(x\right)\right|\\ =\left|f\right|\end{array}$

Using composite of continuous function, $g\left(f\left(x\right)\right)$ is also continuous.

Hence, $\left|f\right|$ is continuous.