Does there exist a function satisfying certain conditions? Does there exists a function $f : \mathbb{R} → \mathbb{R}$ satisfying, $f(−1) = −1$, $f(1) = 1$ and $|f(x)−f(y)|\leq |x−y|^{3/2}$ , for all $x, y \in \mathbb{R}$.
First, I assumed that such a function exist. Then by using mean value theorem, I came to the conclusion that $f(0)=0$. After that what should I do? I want to get a contradiction as the answer to the question is NO.
Thanks!

By assumption
$$\left|\dfrac{f(x)-f(y)}{x-y}\right|\leq |x-y|^{1/2}$$
which shows that
$$\lim_{y\rightarrow x}\dfrac{f(x)-f(y)}{x-y}=0$$
so that $f$ is everywhere differentiable and $f'(x)=0$. This is inconsistent with $f(-1)\not=f(1)$.
