Let be a dense subset of . Suppose that is integrable on and for all . Prove that .
So every point in is either is in or a limit point of . So define .
From here how would we show that ?
Show that, for any dissection of the interval [a,b], the lower Riemann sum for f is ≤0 and the upper Riemann sum is ≥0. It follows that if f is integrable then its integral is 0.