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**ABigSmile** $\displaystyle f $ is differentiable on $\displaystyle [a,b]$. Prove if $\displaystyle f'$ is increasing on $\displaystyle (a,b)$, then $\displaystyle f'$ is cont. on $\displaystyle (a,b)$.

Ok since $\displaystyle f$ is diff. on $\displaystyle [a,b]$ it's also continuous on $\displaystyle [a,b]$. Since it's increasing we have $\displaystyle f'(x) \ge 0 $ for any $\displaystyle x \in (a,b)$. The first two satisfy the MVT for derivatives so how do I use the MVT and the fact that $\displaystyle f'$ is increasing to show it's derivative is continous? I struggle implementing MVT into my proofs apparently.