Math 2.0

statement theorem

Continuous functions are Riemann integrable

Statement

Let a<ba < b and let f ⁣:[a,b]→Rf \colon [a, b] \to \mathbb{R} be continuous. Then ff is Riemann integrable on [a,b][a, b].

Remarks

By the Heine–Cantor theorem, ff is uniformly continuous. Choose δ\delta for ε/(b−a)\varepsilon/(b - a). For a partition of mesh less than δ\delta, U(f,P)−L(f,P)<εU(f, P) - L(f, P) < \varepsilon. Then Riemann's criterion applies. More generally, a bounded function with finitely many discontinuities is Riemann integrable.

Explanations

L1 human

On short pieces of the interval, a continuous function changes very little. So the upper and lower sums are close for fine partitions.

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To read this node, read these 28 nodes in this order. The foundations come first, and the list ends with this node.

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 sequencedefinitionSubsequence, definition, no evidence levelSubsequencedefinitionContinuity 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 sequencedefinitionMonotone sequence, definition, no evidence levelMonotone sequencedefinitionContinuous function, definition, no evidence levelContinuous functiondefinitionEpsilon-delta definition of continuity, statement, no evidence levelEpsilon-delta definit…statementRiemann integral, definition, no evidence levelRiemann integraldefinitionUpper and lower Darboux sums, definition, no evidence levelUpper and lower Darbo…definitionConvergent sequence, definition, no evidence levelConvergent sequencedefinitionMonotone convergence theorem for sequences, statement, no evidence levelMonotone convergence …statementEvery sequence has a monotone subsequence, statement, no evidence levelEvery sequence has a …statementUniform continuity, definition, no evidence levelUniform continuitydefinitionUpper and lower Darboux integrals, definition, no evidence levelUpper and lower Darbo…definitionBolzano–Weierstrass theorem, statement, no evidence levelBolzano–Weierstrass t…statementUniform continuity on a set, definition, no evidence levelUniform continuity on…definitionDarboux and Riemann integrals agree, statement, no evidence levelDarboux and Riemann i…statementContinuous functions on a closed bounded interval are uniformly continuous, statement, no evidence levelContinuous functions …statementRiemann's criterion for integrability, statement, no evidence levelRiemann's criterion f…statementContinuous functions are Riemann integrable, statement, no evidence levelContinuous functions …statement
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. Subsequence definition
  9. Continuity at a point definition
  10. Riemann integrable function definition
  11. Bounded sequence definition
  12. Limit of a sequence definition
  13. Monotone sequence definition
  14. Continuous function definition
  15. Epsilon-delta definition of continuity statement
  16. Riemann integral definition
  17. Upper and lower Darboux sums definition
  18. Convergent sequence definition
  19. Monotone convergence theorem for sequences statement
  20. Every sequence has a monotone subsequence statement
  21. Uniform continuity definition
  22. Upper and lower Darboux integrals definition
  23. Bolzano–Weierstrass theorem statement
  24. Uniform continuity on a set definition
  25. Darboux and Riemann integrals agree statement
  26. Continuous functions on a closed bounded interval are uniformly continuous statement
  27. Riemann's criterion for integrability statement
  28. Continuous functions are Riemann integrable statement

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Sources

  1. Web https://en.wikipedia.org/wiki/Riemann_integral, Integrability

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