1. ## Rolle's Theorem

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?

2. Originally Posted by 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?
I don't understand the question? Does $\displaystyle f$ have to be continuous on $\displaystyle [a,b]$ otherwise this follows directly since any continuous function on a compact space is uniformly continuous.

3. Originally Posted by ABigSmile
"if $\displaystyle f$ is differentiable on $\displaystyle (a,b)$, and $\displaystyle f(a) = f(b) = 0$, then $\displaystyle f$ is uniformly continuous on $\displaystyle [a,b]$"
The problem I am trying to solve is this. It doesn't say prove it so I am assuming it could be a true or false statement. Also I am pretty sure $\displaystyle f$ does not have to be continuous on $\displaystyle [a,b]$ .Does that clear anything up? Sorry for the confusion.

4. Originally Posted by ABigSmile
The problem I am trying to solve is this. It doesn't say prove it so I am assuming it could be a true or false statement. Also I am pretty sure $\displaystyle f$ does not have to be continuous on $\displaystyle [a,b]$ .Does that clear anything up? Sorry for the confusion.
Ok. So why doesn't the function $\displaystyle f(x)=\begin{cases} 0 & \mbox{if} \quad x=0,1\\ 1 & \mbox{if} \quad 0<x<1\end{cases}$ serve as a counter example?. Clearly $\displaystyle f(a)=f(b)=0$ and $\displaystyle f$ is differentiable on $\displaystyle (0,1)$ but it isn't continuous, let alone uniformly continuous, on $\displaystyle [0,1]$.