Let a compact interval $[a,b]\subset \mathbb{R}$. If a function $f:[a,b]\to \mathbb{R}$ is continuous and increasing on $[a,b]$, then what can be said about the derivative $f'$? Is it continuous? If it is, then how do we prove it? I came up with this claim, because I want to prove that if $f$ is continuous and increasing on $[a,b]$, then it its derivative is bounded on $[a,b]$.
This is not a homework. I just wanted to fix it. To be honest, I studied an article on Henstock-Stieltjes integral and the problem that I posted is one of the stated statements in that article.
Thanks in advance...