Math 2.0

Real functions Theorem

First fundamental theorem of calculus

Builds on Mean value theorem for integralsAdditivity of the integral over intervalsDerivative

Statement

Let a<ba < b and let f ⁣:[a,b]→ℝf \colon [a, b] \to \mathbb{R} be Riemann integrable. Define F(x)=∫axf(t) dtF(x) = \int_a^x f(t) \, dt for x∈[a,b]x \in [a, b]. Then FF is continuous on [a,b][a, b]. If ff is continuous at c∈(a,b)c \in (a, b), then FF is differentiable at cc and F′(c)=f(c)F'(c) = f(c).

Remarks

If ∣f∣≤M|f| \le M, then ∣F(y)−F(x)∣≤M∣y−x∣|F(y) - F(x)| \le M |y - x|. So FF is Lipschitz continuous. By additivity, F(c+h)−F(c)h\frac{F(c + h) - F(c)}{h} is the average of ff over the interval between cc and c+hc + h. Continuity at cc makes this average tend to f(c)f(c). Mathlib states a strict form as intervalIntegral.integral_hasStrictDerivAt_of_tendsto_ae_right.

Reading path

To understand this page from the ground up, read these 28 pages in order. The foundations come first.

Show the path as a graph
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 functiondefinitionRiemann integral, definition, no evidence levelRiemann integraldefinitionConvergent 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…statementDefinite integral, definition, no evidence levelDefinite integraldefinitionBolzano–Weierstrass theorem, statement, no evidence levelBolzano–Weierstrass t…statementAdditivity 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…statementExtreme value theorem, statement, no evidence levelExtreme value theoremstatementMean value theorem for integrals, statement, no evidence levelMean value theorem fo…statementFirst fundamental theorem of calculus, statement, no evidence levelFirst fundamental the…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. Riemann integral definition
  17. Convergent sequence definition
  18. Monotone convergence theorem for sequences statement
  19. Every sequence has a monotone subsequence statement
  20. Intermediate value theorem statement
  21. Definite integral definition
  22. Bolzano–Weierstrass theorem statement
  23. Additivity of the integral over intervals statement
  24. Boundedness theorem statement
  25. Monotonicity of the integral statement
  26. Extreme value theorem statement
  27. Mean value theorem for integrals statement
  28. First fundamental theorem of calculus statement

Where this leads

15 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Fundamental_theorem_of_calculus, First part
About this page
Tag063H5KJC69
URIm2://063H5KJC69
Kindstatement, theorem
WritingW1 stated
MSC26A42
StatusThis copy of the site has no build data. Run m2 status 063H5KJC69.
UsesMean value theorem for integrals, Additivity of the integral over intervals, Derivative

Markdown · JSON · Edit on GitHub

What this page needs for W2

  • edge about out: the concepts that the statement is about (0 of 1)
  • edge proves in: at least one proof node (0 of 1)
  • edge instance_of or illustrates in: one example that shows the statement at work (0 of 1)
  • explanation L2 (0 of 1)

The writing levels say how complete the text of a page is. They do not measure its truth.

What links here

Used by