you have only one attempt for this question. it is true that if $f$ is a continuous on an interval, then so…

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.

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.

Answer

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