How to show that every $\alpha$-Hölder function, with $\alpha>1$, is constant? Suppose $f:(a,b) \to \mathbb{R} $ satisfy $|f(x) - f(y) | \le M |x-y|^\alpha$ for some  $\alpha >1$
and all $x,y \in (a,b) $. Prove that $f$ is constant on $(a,b)$. 
I'm not sure which theorem should I look to prove this question. Can you guys give me a bit of hint? First of all how to prove some function $f(x)$ is constant on $(a,b)$? Just show $f'(x) = 0$?

Hint: Show that $f'(y)$ exists and is equal to $0$ for all $y$. Then as usual by the Mean Value Theorem our function is constant. 
