Math 2.0

Real functions Theorem

Darboux and Riemann integrals agree

Builds on Upper and lower Darboux integralsRiemann integrable functionRiemann integral

Statement

Let a<ba < b and let f ⁣:[a,b]→ℝf \colon [a, b] \to \mathbb{R} be bounded. Then ff is Riemann integrable if and only if its upper and lower Darboux integrals are equal. In this case the common value is the Riemann integral ∫abf(x) dx\int_a^b f(x)\, dx.

Remarks

Each Riemann sum lies between the lower and upper Darboux sums. This gives one direction. For the other direction, one shows that Darboux sums for partitions of small mesh are close to the Darboux integrals. So the Riemann and the Darboux approaches define the same integral.

Reading path

To understand this page from the ground up, read these 11 pages in order. The foundations come first.

Show the path as a graph
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…statement
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
  • no evidence level
  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

Where this leads

10 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Darboux_integral, Refinement of a partition and relation to Riemann integration
About this page
Tag021XJAQVDC
URIm2://021XJAQVDC
Kindstatement, theorem
WritingW0 stub
MSC26A42
StatusThis copy of the site has no build data. Run m2 status 021XJAQVDC.
UsesUpper and lower Darboux integrals, Riemann integrable function, Riemann integral

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