Could someone please check my work

Theorem 5.4.6 Suppose f: 0 - R is continuous on a compact Sot D. Then of is uniformly continuous on D. Therem 5.4.9 A function f: (ab)- R is uniformly continuous on (a, 6) iff it can be extended To a funcrit f That is continuous on [ogb]. 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 , letx = and y = a + = for some positive integer, n. Then, if x, y E (0, 2) and |x - yl = = 0 and choose 8 = - . Then, if x, y E [0, 4] and |x - y)