Question

Assertion :$f\left(x\right)=\left|\left[x\right]x\right|$ in $xϵ[-1,2]$ where $\left[\right]$ represents greatest integer function is non differentiable at $x=2$ Reason: Discontinuous function is always non differentiable

• 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

$f\left(x\right)=\left|\left[x\right]x\right|$

at $x=2$

as $x\to 2^{-}\left[x\right]=1$

$\underset{x\to 2^{-}}{lim}f\left(x\right)=\underset{x\to 2^{-}}{lim}|1\cdot x|=2$

as $x\to 2^{-}\left[x\right]=2$

$\underset{x\to 2^{+}}{lim}f\left(x\right)=\underset{x\to 2^{+}}{lim}|2\cdot x|=2\cdot 2=4$

LHL$\ne$RHL

It is discontinuous at $2$

$\to$ It is not differentiable.