Lecture 21 (Real Analysis 2020)

Lemma 6.21 says: if f is integrable and a number A satisfies L(f, p) ≤ A ≤ U(f, p) for every partition p, then:

1. A.A equals the gap of every partition
2. B.A is the supremum of all lower sums only
3. C.f is continuous on [a, b]
4. D.the integral of f over [a, b] equals A