Let $u$ be a bounded function and let a closed set $E$ be given. The compliment of $E$ can be covered with a Whitney type covering $B_i$ such that the following are satisfied:

1) $E^c \subset \bigcup 4B_i$

2) $16B_i \cap E \neq \emptyset$

3) $\sum \chi_{4B_i}(x) \leq C(n)$

4) If $r_i$ is the radius of $B_i$, and if $B_i \cap B_j \neq \emptyset$, then $r_i \leq 2 r_j \leq 4r_i$.

The rest of the properties of the standard Whitney covering also holds.

I have the following uniform $C^{\alpha}$ bound: $$|u(x) - u(y)| \leq C |x-y|^{\alpha}$$ for all $x,y \in 8B_i$ with the constant $C$ and $\alpha$ being independent of the Whitney covering.

Question: How do I show that my function $u$ is Hölder continuous on $E^c$ with a uniform bound?