Is there exists a function $f$ such that $f(-1)=-1$ and $f(1)=1$ and $|f(x)-f(y)| \leq |x-y|^{3/2}$ for all $x,y \in \mathbb{R}$
How to proceed ? Nothing else is mentioned about the function. I have taken a course on real analysis ,but this type of question still bothers me . Please help.
Note that when $x\not = y$, we have that $|\frac{f(x)-f(y)}{x-y}|\le \sqrt{|x-y|}$. If we take the limit as $y\to x$ on both sides, we get information about the derivative of $f$ at any point $x$.