I'm not quite sure how to do this. Here is the definition of uniform continuity (which my professor stipulated we must use to prove this):
$$\forall\epsilon >0, \exists\delta > 0, \forall x,y\in (0, 1), |x-y|<\delta \Rightarrow |f(x)-f(y)| <\epsilon$$
So, I know I must fix an arbitrary epsilon... how do I choose the delta? Forgive me for not having much to start with, my book (Springer) doesn't have concrete examples of uniform continuity proofs to go off of.