Question

you have only one attempt for this question. it is true that if $f$ is a continuous on an interval, then so is $|f|$. but is it true that if $|f|$ is continuous then so is $f$? answer yes for yes or no for no.

Explanation:

Step1: Consider a counter - example

Let \(f(x)=

$$\begin{cases}1, & x\geq0\\ - 1, & x < 0\end{cases}$$

\).

Step2: Analyze \(|f(x)|\)

\(|f(x)| = 1\) for all \(x\in\mathbb{R}\). So \(|f(x)|\) is continuous on \(\mathbb{R}\).

Step3: Analyze \(f(x)\)

\(f(x)\) has a jump discontinuity at \(x = 0\).

Answer:

no