I would like to prove the following inequality:
Let $f$ be a function $f:\mathbb R\to\mathbb R$ such that $0\leq f\leq 1$ on $\mathbb R$ and $f=1$ on $[1,+\infty)$ and $f=0$ on $(-\infty,-1]$. Prove that for $s>0 $, $$ \frac{1}{s}\leq \left| A_s \right| $$ where $A_s$ is a set of $(x,y)\in\mathbb R^2$ so that $\frac{|f(x)-f(y)|}{|x-y|^2}>s$.
If $s$ is big, I think then $A$ contain a small subset so that $x<-1$ and $y>1$. Since $s$ big then such subset is not empty. We can use the Fubini theorem to get the inequality. However, when $s$ is small, I do not know how to prove it. Any idea to prove this inequality?