**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 .

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I would maybe show that if a continuous function is >0 on an interval of positive length then the integral is positive. I’m not sure if that is something that can be used without proof.

Or, I would just do it explicitly for this problem. So instead of doing 0<f(x)<e+m, I would show that 0<k0.

But you definitely have the main idea of the proof. I just have no idea what they take points off for, and if they would dock you because they actually wanted you to show the integral was >0 explicitly.

Hrm, part of my message was chopped off. 2nd paragraph should say,

0< k< f (x) therefore, the integral of f on the interval from 1/2-d to 1/2d is 2dk.

Man, there should be an edit button for comments, lol…

so the integral is greater than (not equal to necessarily) 2dk.