 Solution

Take any $a<u<v<b$. By differentiability implies continuity, $f$ is continuous on $[u,v]$, and it is differentiable on $(u,v)$. The mean value theorem gives $c\in (u,v)$ for whichTherefore $\boxed{f(u)\le f(v)\text{ whenever }u<v}$: $f$ is a nondecreasing function, which is the non-strict meaning of increasing here.