calculus - Can the derivative of a function be such that it is not continuous?

My guess is that all derivatives ought to be continuous but perhaps there's some obscure example of a function for which this is not the case. Is there any theorem that answers this perhaps?

Answer

The standard counterexample is $f(x)=x^2\sin(\frac{1}{x})$ for $x\neq 0$, with $f(0)=0$.

This function is everywhere differentiable, with $f^{\prime}(x)=2x\sin(\frac{1}{x})-\cos(\frac{1}{x})$ if $x\neq 0$ and $f^{\prime}(0)=0$. However, $f^{\prime}$ is not continuous at zero because $\lim_{x\to0}\cos(\frac{1}{x})$ does not exist.

While $f^{\prime}$ need not be continuous, it does satisfy the intermediate value property. This is known as Darboux's theorem.