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Need solution for RD Sharma maths class 12 chapter 9 Differentiability exercise Very short answer type question  3

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Answer: No

Explanation: Every continuous function is not differentiable.

For example,f(x)=|x|=\left\{\begin{array}{ll} x & x>0 \\ -x & x<0 \end{array}\right\}is continuous at x=0.

But at x=0,

\begin{aligned} L H D &=\lim _{x \rightarrow 0^{-}} \frac{f(x)-f(0)}{x-0} \\\\ &=\lim _{x \rightarrow 0^{-}} \frac{-x-0}{x-0}=\lim _{x \rightarrow 0}-1 \\\\ &=-1 \end{aligned}

\begin{aligned} R H D &=\lim _{x \rightarrow 0^{+}} \frac{f(x)-f(0)}{x-0} \\\\ &=\lim _{x \rightarrow 0^{-}} \frac{x-0}{x-0}=\lim _{x \rightarrow 0} 1 \\\\ &=1 \end{aligned}

As L H D \neq R H D

Therefore, \left | x \right | is not differentiable at x=0.

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