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Darboux sum

For a bounded function,
$$
\overline{\int_a^b}f=\inf_PU(f,P),
\qquad
\underline{\int_a^b}f=\sup_PL(f,P).
$$
The function is Riemann integrable exactly when these values agree.


Upper and lower Darboux integrals

Past exam of the mathematics course of the University of Cambridge / 2009 / ia / Paper 1 / 12E / a / Solution

Past exam of the mathematics course of the University of Cambridge / 2017 / ia / Paper 1 / 12E / Solution

Past exam of the mathematics course of the University of Cambridge / 2024 / ia / Paper 1 / 12D / a / i / Solution

Riemann-integrable function

Upper and lower Darboux integrals

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Upper and lower Darboux integrals

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