# Oscillation and Holder continuity

Where can I find a proof of the follwing fact?

If $$w(u,x_{0},r)=\sup _{B_{r}(x_{0})}u-\inf _{B_{r}(x_{0})}u$$ for some function $$u(x)$$ satisfies $$w\left(u,x_{0},{\tfrac {r}{2}}\right)\leq \lambda w\left(u,x_{0},r\right)$$ for a fixed $$0 < \lambda < 1$$ and all sufficiently small values of $$r$$, then $$u$$ is Holder continuous.

Just prove it yourself:

Take $$r=1$$. Then $$w(x_0,2^{-n})\leq \lambda^n w(x_0,1)=:C\lambda^n.$$ To estimate $$|u(x_0)-u(y)|$$, where $$y$$ is close to $$x_0$$, choose $$n$$ so that $$|x_0-y|\in[2^{-n-1},2^{-n}].$$ Then $$|u(x_0)-u(y)|\leq w(x_0,2^{-n})\leq C\lambda^{n}=C. 2^{-hn}\leq 2^hC_1|x_0-y|^h,$$ where $$h=-\log\lambda/\log2$$.