Math 2.0

counterexample

The Dirichlet function is not Riemann integrable

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.

Explanations

L1 human

Every Riemann sum of this function is 11 or 00 or in between, depending only on the choice of tags. So the sums cannot settle down to one number.

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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 node uses the nodes above it. The colour of the bar on a box shows the evidence level, and the box also gives it as text. The list below has the same nodes.
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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

Sources

  1. Web https://en.wikipedia.org/wiki/Dirichlet_function, Integration properties

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