Question

Identify the correct statement-

• A. If $f\left(x\right)$ is differentiable at $x=a$, $\left|f\right(x\left)\right|$ will also be differentiable at $x=a$
• B. If $f\left(x\right)$ is continuous at $x=a$, $\left|f\right(x\left)\right|$ will also be continuous at $x=a$
• C. If $f\left(x\right)$ is discontinuous at $x=a$, $\left|f\right(x\left)\right|$ will also be discontinuous at $x=a$
• D. If $\left|f\right(x\left)\right|$ is continuous at $x=a$, $f\left(x\right)$ too will be continuous at $x=a$

Answer 1

The correct option is B If $f\left(x\right)$ is continuous at $x=a$, $\left|f\right(x\left)\right|$ will also be continuous at $x=a$

If $f\left(x\right)$ is continuous at $x=a$, then $|f\left(x\right)|$ will surely be continuous at $x=a.$
We give counter examples for the other three options.
If $f\left(x\right)$ is differentiable, $|f\left(x\right)|$ need not be since $|x|$ is an example at $x=0.$
Also, we can have a discontinuous function, having values $-h$ at $x=a^{-}$ and $h$ at $x=a^{+}$. This function becomes continuous when taken modulus.