The function $f$ is $f:\mathbb{R}\rightarrow\mathbb{R}$ and $0<\alpha<1$. I need to show that the set of all $x\in\mathbb{R}$ such that $\displaystyle{\lim_{r \to 0}}\sup\{\frac{\mid f(u)-f(v)\mid}{\mid u-v\mid^\alpha}:u\neq v,\mid u-x\mid <r,\mid v-x\mid <r\}=0$
is a Borel set.
From what I know, if I can write the set as a union of closed sets or intersection of open sets, the $G_\delta$ or $F_{\sigma}$ then I am done.
Also, correct me if I am wrong, the set of $x$ above is the set of all the stationary points. I only know how to change the limit into intersection
$\underset{n}{\cap}\sup\{\frac{\mid f(u)-f(v)\mid}{\mid u-v\mid^\alpha}:u\neq v,\mid u-x\mid <\frac{1}{n},\mid v-x\mid <\frac{1}{n}\}$
How to settle the equal to zero and the sup? Or there is another way to show that this is a Borel set? Am I in the right direction?