Question

11.6. Properties of Riemann integrable functions.​

Answer 1

Answer:

A bounded function f:[a,b]→R is Riemann integrable if and only if ∀ϵ>0,∃Qsuch thatU(Q,f)−L(Q,f)<ϵ. Proof. If f is Riemann integrable, then for all ϵ>0 there exists P1,P2 such that U(P2,f)−∫fdx<ϵ/2 and ∫fdx−L(P1,f)<ϵ/2.