Math 2.0

statement theorem

Riemann's criterion for integrability

Statement

Let a<ba < b and let f ⁣:[a,b]→Rf \colon [a, b] \to \mathbb{R} be bounded. Then ff is Riemann integrable on [a,b][a, b] if and only if the following holds. For each ε>0\varepsilon > 0 there is a partition PP of [a,b][a, b] with U(f,P)−L(f,P)<εU(f, P) - L(f, P) < \varepsilon.

Remarks

Here UU and LL are the upper and lower Darboux sums. The condition says that the upper and lower Darboux integrals are equal. So it follows from the equality of Darboux and Riemann integrability. This is the usual tool to prove that continuous and monotone functions are integrable.

Explanations

L1 human

A function is integrable when we can trap its graph between step functions whose areas differ by as little as we like.

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Partition of an interval, definition, no evidence levelPartition of an inter…definitionSequence, definition, no evidence levelSequencedefinitionMesh of a partition, definition, no evidence levelMesh of a partitiondefinitionRiemann sum, definition, no evidence levelRiemann sumdefinitionReal sequence, definition, no evidence levelReal sequencedefinitionRiemann integrable function, definition, no evidence levelRiemann integrable fu…definitionBounded sequence, definition, no evidence levelBounded sequencedefinitionRiemann integral, definition, no evidence levelRiemann integraldefinitionUpper and lower Darboux sums, definition, no evidence levelUpper and lower Darbo…definitionUpper and lower Darboux integrals, definition, no evidence levelUpper and lower Darbo…definitionDarboux and Riemann integrals agree, statement, no evidence levelDarboux and Riemann i…statementRiemann's criterion for integrability, statement, no evidence levelRiemann's criterion f…statement
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  1. Partition of an interval definition
  2. Sequence definition
  3. Mesh of a partition definition
  4. Riemann sum definition
  5. Real sequence definition
  6. Riemann integrable function definition
  7. Bounded sequence definition
  8. Riemann integral definition
  9. Upper and lower Darboux sums definition
  10. Upper and lower Darboux integrals definition
  11. Darboux and Riemann integrals agree statement
  12. Riemann's criterion for integrability statement

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Sources

  1. Web https://en.wikipedia.org/wiki/Darboux_integral, Refinement of a partition and relation to Riemann integration

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