This is not exactly Rolle's Theorem but it's a problem that was given to me that is similar that is bugging me. It states "if is differentiable on , and , then is uniformly continuous on " I understand since is differentiable on that it must also be continuous on .

But continuity doesn't imply uniform continuity. So is there any way to show uniform continuity? Because if there is, I haven't been able to figure it out so far. Is this statement always false since our interval is open? Would providing a simple counter example suffice if that's the case?