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?