◀ ▲ ▶Branches / Analysis / Proposition: A Necessary and a Sufficient Condition for Riemann Integrable Functions
The above example calls for a sufficient condition, under which a function is Riemann-integrable. The following proposition provides such a condition.
Proposition: A Necessary and a Sufficient Condition for Riemann Integrable Functions
Let \([a,b]\) be a closed real interval. A function \(f:[a,b]\mapsto\mathbb R\) is Riemann integrable, if and only if
- \(f\) is bounded, and
- for every \(\epsilon > 0\) there exist some step functions \(\phi:[a,b]\mapsto\mathbb R\) and \(\psi:[a,b]\mapsto\mathbb R\) with \[\phi\le f\le \psi\] and \[\int_a^b\psi(x)dx-\int_a^b\phi(x)dx\le \epsilon.\]
Table of Contents
Proofs: 1
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Proofs: 1 2
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References
Bibliography
- Forster Otto: "Analysis 1, Differential- und Integralrechnung einer Veränderlichen", Vieweg Studium, 1983
