Math 2.0

Real functions Counterexample

The Dirichlet function is not Riemann integrable

Builds on Dirichlet functionUpper and lower Darboux integralsDarboux and Riemann integrals agree

Counterexample

Let f ⁣:[0,1]→ℝf \colon [0, 1] \to \mathbb{R} be the Dirichlet function: f(x)=1f(x) = 1 for rational xx and f(x)=0f(x) = 0 for irrational xx. Each subinterval of positive length contains both rational and irrational numbers. So for each partition PP, U(f,P)=1U(f, P) = 1 and L(f,P)=0L(f, P) = 0. The upper Darboux integral is 11 and the lower one is 00. So ff is not Riemann integrable.

Remarks

The function is bounded, so the failure is not caused by unboundedness. The function is Lebesgue integrable, with Lebesgue integral 00, since ℚ\mathbb{Q} has measure zero. It is the pointwise limit of Riemann integrable functions. So pointwise limits do not keep Riemann integrability.

Reading path

To understand this page from the ground up, read these 18 pages in order. The foundations come first.

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Limit point of a set, definition, no evidence levelLimit point of a setdefinitionPartition of an interval, definition, no evidence levelPartition of an inter…definitionSequence, definition, no evidence levelSequencedefinitionLimit of a function, definition, no evidence levelLimit of a functiondefinitionMesh of a partition, definition, no evidence levelMesh of a partitiondefinitionRiemann sum, definition, no evidence levelRiemann sumdefinitionReal sequence, definition, no evidence levelReal sequencedefinitionContinuity at a point, definition, no evidence levelContinuity at a pointdefinitionRiemann integrable function, definition, no evidence levelRiemann integrable fu…definitionBounded sequence, definition, no evidence levelBounded sequencedefinitionLimit of a sequence, definition, no evidence levelLimit of a sequencedefinitionRiemann integral, definition, no evidence levelRiemann integraldefinitionUpper and lower Darboux sums, definition, no evidence levelUpper and lower Darbo…definitionSequential continuity, definition, no evidence levelSequential continuitydefinitionDirichlet function, counterexample, no evidence levelDirichlet functioncounterexampleUpper and lower Darboux integrals, definition, no evidence levelUpper and lower Darbo…definitionDarboux and Riemann integrals agree, statement, no evidence levelDarboux and Riemann i…statementThe Dirichlet function is not Riemann integrable, counterexample, no evidence levelThe Dirichlet functio…counterexample
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.
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  1. Limit point of a set definition
  2. Partition of an interval definition
  3. Sequence definition
  4. Limit of a function definition
  5. Mesh of a partition definition
  6. Riemann sum definition
  7. Real sequence definition
  8. Continuity at a point definition
  9. Riemann integrable function definition
  10. Bounded sequence definition
  11. Limit of a sequence definition
  12. Riemann integral definition
  13. Upper and lower Darboux sums definition
  14. Sequential continuity definition
  15. Dirichlet function counterexample
  16. Upper and lower Darboux integrals definition
  17. Darboux and Riemann integrals agree statement
  18. The Dirichlet function is not Riemann integrable counterexample

Further reading

  1. Web https://en.wikipedia.org/wiki/Dirichlet_function, Integration properties
About this page
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Kindcounterexample
WritingW0 stub
MSC26A42
UsesDirichlet function, Upper and lower Darboux integrals, Darboux and Riemann integrals agree

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