Can you give an example of a function $f(x)$ which is differentiable but not continuously differentiable and there exists a nbd $N$ around a point $c$ such that $f'(x) >(<) 0 \forall x \in N $ and $f'(x)$ is not continuous at the point $c$?
I am basically trying to search a function which is monotone in an open interval but it is not continuously differentiable at that interval.
Can anyone please help me to find that kind of a function?