Abstract
We know that if f is integrable, then the lower and upper sums of every partition F approximate its integral from below and above, and so the difference between either sum and the integral is at most \(S_{F} - s_{F} =\varOmega _{F}\), the oscillatory sum corresponding to F.
Thus the oscillatory sum is an upper bound for the difference between the approximating sums and the integral.
We also know that if f is integrable, then the oscillating sum can become smaller than any fixed positive number for a sufficiently fine partition (see Theorem 14.23).
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Laczkovich, M., Sós, V.T. (2015). Functions of Bounded Variation. In: Real Analysis. Undergraduate Texts in Mathematics. Springer, New York, NY. https://doi.org/10.1007/978-1-4939-2766-1_17
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DOI: https://doi.org/10.1007/978-1-4939-2766-1_17
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