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This is my question :

Let $f$ be defined on an interval $I$, and suppose there exists an $M>0$ and $\alpha>0$ such that $$ |f(x) - f(y)| \leq M|x -y|^\alpha, $$ for $x,y \in I$. Prove that $f$ is uniformly continuous on $I$. If $\alpha>1$, prove that $f$ is constant on $I$.

Should I use the mean value theorem for this problem?

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    How do you hope to use the mean value theorem? Where are you going to get a differentiable function from?2012-11-22
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    For the first part: Given $\epsilon>0$, find a $\delta>0$ such that $M\delta^\alpha<\epsilon$ and see if it helps.2012-11-22
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    For the second part: I'll just interpret Chris Eagle's comment as I see it: There is a differentiable function here. Why is it differentiable?2012-11-22

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