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Q.
If $f$ is an even function such that $\displaystyle\lim _{h \rightarrow 0} \frac{f(h)-f(0)}{h}$ has some finite non-zero value, then
Continuity and Differentiability
Solution:
Let $f'\left(0^{+}\right)=\displaystyle\lim _{h \rightarrow 0} \frac{f(h)-f(0)}{h}=k$ (say)
$\therefore f'\left(0^{-}\right)=\displaystyle\lim _{h \rightarrow 0} \frac{f(0)-f(0-h)}{h}$
$=\displaystyle\lim _{h \rightarrow 0} \frac{f(0)-f(h)}{h}=-k$
Since $f'\left(0^{+}\right) \neq f'\left(0^{-}\right)$, but both are finite, we can say that $f(x)$ is continuous at $x=0$ but not differentiable at $x=0 $