Could someone please check my work

Please state all definitions and theorems that you will need; Definition 5.4.1 Let f : D - R . We say that f is uniformly continuous on D if for every & > 0 there exists a 6 > 0 such that If(a) - f(y) | 0 and choose 61 > 0 . Then, if x, y E D and (x - y| 0 . Then, if x, y E D and |x - y|