Derivative of differentiable function need not be continuous
From Calculus
Statement
It is possible to have a function defined for real numbers such that
is a differentiable function everywhere on its domain but the derivative
is not a continuous function.
Equivalently, a differentiable function on the real numbers need not be a continuously differentiable function.
Proof
Example with an isolated discontinuity
Consider the function:
Then, we have:
In particular, we note that but
does not exist. Thus,
is not a continuous function at 0.
For details, see square times sine of reciprocal function#First derivative.
The video below covers this example.