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Question 8.T.5: The following statements are equivalent: (i) f ∈ R(a, b) wit...

The following statements are equivalent:

(i) f ∈ \mathcal{R}(a, b) with integral over [a, b] equal to A.

(ii) For any ε > 0, there is a δ > 0 such that, if P is any partition satisfying \left\|P\right\| < δ and α is any mark on P, then

|S(f, P, α) − A| < ε,

that is,

\underset{\left\|P\right\|→0 }{\lim} S (f, P, α) = A.

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