Problem Statement: Using properties of the Riemann Integral show that if is Riemann Integrable on and then is uniformly continuous on .
Proof: We wish to show that for every there exists a such that for every in it follows that .
Since is Riemann Integrable on it follows that is continuous on a compact set and thus bounded. Let such that for every .
Let and . Consider such that . Wlog assume that .
Therefore, is uniformly continuous on .
Reflection: Again, another one that really falls out straight by definition. The reason we’re able to combine the integrals is because the is fixed in . Also, we are given that is Riemann Integrable and so that tells us that it’s bounded, which is really where the meat of the proof lies.