Question

Assertion :If $\left|f\right(x\left)\right|\le \left|x\right|$ for all $x\in R$ then $\left|f\right|$ is continuous at $x=0$ Reason: If f is continuous then $\left|f\right|$ is continuous

• A. Both Assertion and Reason are correct and Reason is the correct explanation for Assertion
• B. Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion
• C. Assertion is correct but Reason is incorrect
• D. Both Assertion and Reason are incorrect

Answer 1

The correct option is A Both Assertion and Reason are correct and Reason is the correct explanation for Assertion

Put $x=0$ in $|f\left(x\right)|\le |x|$ so $f\left(0\right)\le 0$Thus $|f\left(0\right)|=0\Rightarrow f\left(0\right)=0$. We can write
$|f\left(x\right)|\le |x|$ as $|f\left(x\right)-f\left(0\right)|\le |x-0|$
Thus f is continuous at 0. So applying statement-2 $|f|$ is continuous at $x=0$. Of course, if f is continuous then $|f|$ is continuous function.