Let $f$ be a continuous function that is differentiable on $[a,b]$. It is known to us $f^\prime$ can be discontinuous, and $f^\prime$ can be unbounded, for instance, consider the function $f(x)=x^{3/2}\sin(1/x)$. Now my question is, how "unbound" can be say about $f^\prime$? To be more general, are there any such $f$ has a derivative that is unbounded on any subinterval $I$ of $[a,b]$?
I tried Baire's category theorem, but the best result I can obtain here is that $f^\prime$ is bounded on some set that is dense in an open interval.