Uniformly Continuous

Let and be Metric spaces. A function is uniformly continuous if

Lemma

If is uniformly continuous then it is Continuous.

Theorem

If is continuous on a closed interval then it is uniformly continuous on .

Proof sketch

Suppose otherwise. Choose a “bad” and , along with . Use BWT to find convergent (which has to stay in the closed interval). Then also . Now find a contradiction with continuity at .

Theorem

If is continuous on then it is Riemann Integrable on .

Proof sketch

By previous theorem, is uniformly continuous. Now choose a dissection s.t. . Write out and and bind them appropriately - done.