Problem Statement: Assume is a bounded Riemann Integrable function on . Set . Prove that is uniformly continuous on .
Proof: Let such that for every . Let and let . Then for it follows that . We are given that is Riemann Integrable on so these integrals exist and we may manipulate them. WLOG assume that , then . Thus is uniformly continuous on .
Reflection: This proof has reminded me to not over think these problems. When I first approached this problem I still had the idea of a bounded derivative implies uniformly continuous fresh in my mind. This led me to use the Fundamental Theorem of Calculus, parts one and two to show that was bounded on and using the MVT we get that is uniformly continuous. This was all fine and dandy until I realized that to use FTC we need to be continuous. Oh no!! We don’t know anything about other than that it’s Riemann Integrable. Unlike being differentiable, which tells us that is continuous, being Riemann Integrable really doesn’t tell us a whole lot…So, after realizing that we are not guaranteed is continuous we had to try another simpler, more direct, approach. So, the moral of the story is, don’t over think!!