Question: I stumbled upon a question involving discontinuous derivative but I couldn't understand how could it happen that the derivative of a function (let's say $f(x)$) is defined at point (let's suppose $c$) but then it is discontinuous (or undefined) at the same point $c$. Is there a problem in calculus?

(The function $f(x)$ is also continuous at $c$)

Answer: 
The function
$$
f(x) = \begin{cases}
x^2\sin(1/x), &\text{if } x\ne0, \\
0, &\text{if } x=0
\end{cases}
$$
is differentiable everywhere (in particular, $f'(0)=0$) but its derivative is not continuous at $x=0$.

Whether this is a "problem" in calculus is in the eye of the beholder! (By way of analogy, the school of Pythagoras discovered that a real number could be defined but could not be written as a quotient of two integers. To them that might have seemed like a problem, but today we're pretty comfortable with the idea of irrational numbers.)

Certainly the drive to formalize these concepts in analysis in the 19th century was motivated in large part by "pathological" examples like this one. In the end, the definitions of continuity and differentiability we settled one do permit functions to have derivatives that are defined everywhere but not continuous everywhere, just as they permit functions that are themselves defined everywhere but not continuous everywhere.

Mathematical definitions develop in order to reflect our intuitions as much as possible within mathematics. But once we have settled on mathematical definitions, if ever their consequences run counter to our intuitions, we must be prepared to update our intuitions.
Remember that it’s entirely possible for a function to be continuous at a point where the derivative is not. Moreover, take care not to confuse continuity with existence (or being well defined for that matter). There is no prerequisite for differentiability of a function based on the differentiability of its derivative. A thorough discussion can be found here: [Discontinuous derivative.](https://math.stackexchange.com/questions/292275/discontinuous-derivative/292380#292380)