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

Question

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 1

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.

Answer 2

The classic family of examples to all questions of this kind is $f_{\alpha}(x)=|x|^\alpha\sin\frac{1}{x}$ for $x\in \mathbb{R}\backslash\{0\}$ and $f_{\alpha}(0)=0$. You choose $\alpha>0$ to ensure that $f$ satifies the properties you want such as $f$ thrice differentiable but the third derivative is not bounded (on any closed interval containing $0$) or continuous or whatever.