Math 2.0

Real functions Corollary

The integral of a continuous function is an antiderivative

Builds on First fundamental theorem of calculusAntiderivativeContinuous functions are Riemann integrable

Statement

Let II be an open interval, let f ⁣:I→ℝf \colon I \to \mathbb{R} be continuous, and let a∈Ia \in I. Then F(x)=∫axf(t) dtF(x) = \int_a^x f(t) \, dt is an antiderivative of ff on II.

Remarks

The integral exists, since continuous functions are integrable. The first fundamental theorem applies at each point of II. So every continuous function on an open interval has an antiderivative. Mathlib states the derivative as Continuous.deriv_integral.

Reading path

To understand this page from the ground up, read these 39 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 sequencedefinitionSubsequence, definition, no evidence levelSubsequencedefinitionContinuity at a point, definition, no evidence levelContinuity at a pointdefinitionDerivative, definition, no evidence levelDerivativedefinitionRiemann 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…statementAntiderivative, definition, no evidence levelAntiderivativedefinitionRiemann 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 …statementIntermediate value theorem, statement, no evidence levelIntermediate value th…statementUniform continuity, definition, no evidence levelUniform continuitydefinitionDefinite integral, definition, no evidence levelDefinite integraldefinitionUpper 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…definitionAdditivity of the integral over intervals, statement, no evidence levelAdditivity of the int…statementBoundedness theorem, statement, no evidence levelBoundedness theoremstatementMonotonicity of the integral, statement, no evidence levelMonotonicity of the i…statementDarboux and Riemann integrals agree, statement, no evidence levelDarboux and Riemann i…statementExtreme value theorem, statement, no evidence levelExtreme value theoremstatementContinuous 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…statementMean value theorem for integrals, statement, no evidence levelMean value theorem fo…statementContinuous functions are Riemann integrable, statement, no evidence levelContinuous functions …statementFirst fundamental theorem of calculus, statement, no evidence levelFirst fundamental the…statementThe integral of a continuous function is an antiderivative, statement, no evidence levelThe integral of a con…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. 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. Derivative definition
  11. Riemann integrable function definition
  12. Bounded sequence definition
  13. Limit of a sequence definition
  14. Monotone sequence definition
  15. Continuous function definition
  16. Epsilon-delta definition of continuity statement
  17. Antiderivative definition
  18. Riemann integral definition
  19. Upper and lower Darboux sums definition
  20. Convergent sequence definition
  21. Monotone convergence theorem for sequences statement
  22. Every sequence has a monotone subsequence statement
  23. Intermediate value theorem statement
  24. Uniform continuity definition
  25. Definite integral definition
  26. Upper and lower Darboux integrals definition
  27. Bolzano–Weierstrass theorem statement
  28. Uniform continuity on a set definition
  29. Additivity of the integral over intervals statement
  30. Boundedness theorem statement
  31. Monotonicity of the integral statement
  32. Darboux and Riemann integrals agree statement
  33. Extreme value theorem statement
  34. Continuous functions on a closed bounded interval are uniformly continuous statement
  35. Riemann's criterion for integrability statement
  36. Mean value theorem for integrals statement
  37. Continuous functions are Riemann integrable statement
  38. First fundamental theorem of calculus statement
  39. The integral of a continuous function is an antiderivative statement

Where this leads

3 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Fundamental_theorem_of_calculus, First part
About this page
Tag0NC2A3HWGT
URIm2://0NC2A3HWGT
Kindstatement, corollary
WritingW0 stub
MSC26A36
StatusThis copy of the site has no build data. Run m2 status 0NC2A3HWGT.
UsesFirst fundamental theorem of calculus, Antiderivative, Continuous functions are Riemann integrable

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