Problem Statement: Suppose that is a continuous real valued function on such that for every and . Show .
Proof: Since is continuous on we know is Riemann Integrable. We are given that and since is continuous we know that for every there exists a such that for every . This implies that for every . This implies that . Thus we may conclude that
Reflection: We know we can even consider the integral since is continuous on the interval. It is the fact the is continuous that really gives us the meat of this proof. Since it’s continuous we know there’s some interval of positive length where since .