Question

Show that 𝑈(−𝑓, 𝑝) = −𝐿(𝑓, 𝑝) and 𝐿(−𝑓, 𝑝) = −𝑈(𝑓, 𝑝)

Answer 1

Suppose that f is integrable on [a; b]. For $\in > 0$ , there exist partitions P₁ and P₂ such that

$L(f, P_1) > L(f) − \frac{\in}{2} \space and \space U(f; P2) < U(f) + \frac{\in}{2}\\ For \space P := P_1 ∪ P_2 \space we \space have\\ L(f) −\frac{\in}{2} < L(f, P_1) ≤ L(f; P ) ≤ U(f, P ) ≤ U(f, P_2) < U(f) + \frac{\in}{2} \\$

Since L(f) = U(f), it follows that $U(f, P ) - L(f,P )< \in$

Implies 𝑈(−𝑓, 𝑝) = −𝐿(𝑓, 𝑝) and 𝐿(−𝑓, 𝑝) = −𝑈(𝑓, 𝑝)