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**ABigSmile** 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 $\displaystyle f $ is differentiable on $\displaystyle (a,b)$, and $\displaystyle f(a) = f(b) = 0$, then $\displaystyle f$ is uniformly continuous on $\displaystyle [a,b]$" I understand since $\displaystyle f$ is differentiable on $\displaystyle (a,b)$ that it must also be continuous on $\displaystyle (a,b)$.

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?