Lecture 20 (Real Analysis 2020)

Lemma 6.17 guarantees that for a continuous f on [a, b] and a partition p there exist points u, v in one partition interval such that:

1. A.0 ≤ U(f, p) − L(f, p) ≤ f(v) − f(u)
2. B.0 ≤ f(v) − f(u) ≤ U(f, p) − L(f, p)
3. C.L(f, p) ≤ f(v) − f(u) ≤ U(f, p)
4. D.U(f, p) − L(f, p) = f(b) − f(a)