Give an example of a function which is continuos but not differentiable at at a point.
Consider a function, $f(x)= \begin{cases}x, & x>0 \\ -x, & x \leq 0\end{cases}$
This mod function is continuous at $x=0$ but not differentiable at $x=0$.
Continuity at $x=0$, we have:
$(\mathrm{LHL}$ at $x=0)$
$\lim _{x \rightarrow 0^{-}} f(x)$
$=\lim _{h \rightarrow 0} f(0-h)$
$=\lim _{h \rightarrow 0}-(0-h)$
$=0$
(RHL at x = 0)
$\lim _{x \rightarrow 0^{+}} f(x)$
$=\lim _{h \rightarrow 0} f(0+h)$
$=\lim _{h \rightarrow 0}(0+h)$
$=0$
and $f(0)=0$
Thus, $\lim _{x \rightarrow 0^{-}} f(x)=\lim _{x \rightarrow 0^{+}} f(x)=f(0)$.
Hence, $f(x)$ is continuous at $x=0$.
Now, we will check the differentiability at x=0, we have:
(LHD at x = 0)
$\lim _{x \rightarrow 0^{-}} \frac{f(x)-f(0)}{x-0}$
$=\lim _{h \rightarrow 0} \frac{f(0-h)-f(0)}{0-h-0}$
$=\lim _{h \rightarrow 0} \frac{-(0-h)-0}{-h}$
$=-1$
(RHD at x = 0)
$\lim _{x \rightarrow 0^{+}} \frac{f(x)-f(0)}{x-0}$
$=\lim _{h \rightarrow 0} \frac{f(0+h)-f(0)}{0+h-0}$
$=\lim _{h \rightarrow 0} \frac{0+h-0}{h}$
$=1$
Thus, $\lim _{x \rightarrow 0^{-}} f(x) \neq \lim _{x \rightarrow 0^{+}} f(x)$
Hence $f(x)$ is not differentiable at $x=0$.
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