Suppose that $f:\Bbb R\to \Bbb R$ is a function satisfying $|f(x)-f(y)|\le |x-y|^\beta $.
If $\beta >0$ show that $f$ is uniformly continuous.
Attempt:
Let $\epsilon>0$ be given. Then choose $\delta =\epsilon^{\dfrac{1}{\beta}}$. So whenever $|x-y|<\delta$
Then $|f(x)-f(y)|\le |x-y|^\beta<\delta^\beta=\epsilon$.
Is the method right ?Please suggest required edits if required.