Question

Assertion :The function f(x)=Ixl is discontinuous at origin. Reason: The function f(x)=Ixl is non differentiable at origin.

• 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. Assertion is incorrect but Reason is correct

Answer 1

The correct option is D Assertion is incorrect but Reason is correct

$f\left(x\right)=\left|x\right|=\{\begin{array}{cc}x& \forall x\ge 0\\ -x& \forall x<0\end{array}$
$\therefore$ $\underset{x\to 0^{+}}{lim}f\left(x\right)=\underset{x\to 0^{-}}{lim}=f\left(0\right)$
$\Rightarrow$ f(x) is continuous at origin
$\therefore$ Assertion (A) is false
Again $f^{\prime }\left(x\right)=\{\begin{array}{cc}1& \forall x\ge 0\\ -1& \forall x<0\end{array}$
$\therefore$ L.H.D. (at x=0)$\ne$R.H.D. (at x=0)
$\Rightarrow$ f(x) is non differentiable at origin.