Suppose and are real valued functions that are Riemann integrable on . Suppose further that for all and that a strict inequality holds for at least one point in . Prove that need not hold. If we add a continuity condition to and , does the above inequality hold?
Proof. We know that . Define and as follows:
Here we have and . Then .
If we had a continuity condition, then I think the above inequality will hold? Do we use the following fact: There exists an open interval in such that and for all ? Is this correct?