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If $f(x)=|x|$, then $f(x)$ is
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Verified Answer
The correct answer is:
Continuous for all x
It is obvious that $|x|$ is continuous for all $x$.
Now,
$\begin{aligned} & \quad R f^{\prime}(x)=\lim _{h \rightarrow 0} \frac{|0+h|-0}{h}=1 \\& \text { Now, } \\
& L f^{\prime}(x)=\lim _{h \rightarrow 0} \frac{|0-h|-0}{-h}=-1\end{aligned}$
Hence $f(x)=|x|$ is not differentiable at $x=0$.
Now,
$\begin{aligned} & \quad R f^{\prime}(x)=\lim _{h \rightarrow 0} \frac{|0+h|-0}{h}=1 \\& \text { Now, } \\
& L f^{\prime}(x)=\lim _{h \rightarrow 0} \frac{|0-h|-0}{-h}=-1\end{aligned}$
Hence $f(x)=|x|$ is not differentiable at $x=0$.
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