Math 2.0

Sequences, series, summability Theorem

Uniform convergence of derivatives allows term-by-term differentiation

Builds on Uniform convergenceMean value theoremUniformly Cauchy sequence of functions

Statement

Let a<ba < b, and let fn ⁣:[a,b]→ℝf_n \colon [a, b] \to \mathbb{R} be differentiable for each nn. Assume that (fn(x0))(f_n(x_0)) converges for some x0∈[a,b]x_0 \in [a, b], and that (fn′)(f_n') converges uniformly on [a,b][a, b] to a function gg. Then (fn)(f_n) converges uniformly on [a,b][a, b] to a differentiable function ff, and f′=gf' = g.

Remarks

The mean value theorem applied to fn−fmf_n - f_m shows that (fn)(f_n) is uniformly Cauchy. Uniform convergence of fnf_n alone is not enough. The functions sin⁡(n2x)n\frac{\sin(n^2 x)}{n} tend to 00 uniformly, but their derivatives ncos⁡(n2x)n \cos(n^2 x) do not converge. Mathlib states a version as hasDerivAt_of_tendstoUniformly.

Reading path

To understand this page from the ground up, read these 30 pages in order. The foundations come first.

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Limit point of a set, definition, no evidence levelLimit point of a setdefinitionLocal maximum, definition, no evidence levelLocal maximumdefinitionSequence, definition, no evidence levelSequencedefinitionLimit of a function, definition, no evidence levelLimit of a functiondefinitionLocal minimum, definition, no evidence levelLocal minimumdefinitionReal sequence, definition, no evidence levelReal sequencedefinitionSubsequence, definition, no evidence levelSubsequencedefinitionContinuity at a point, definition, no evidence levelContinuity at a pointdefinitionDerivative, definition, no evidence levelDerivativedefinitionBounded sequence, definition, no evidence levelBounded sequencedefinitionCauchy sequence, definition, no evidence levelCauchy sequencedefinitionLimit of a sequence, definition, no evidence levelLimit of a sequencedefinitionMonotone sequence, definition, no evidence levelMonotone sequencedefinitionContinuous function, definition, no evidence levelContinuous functiondefinitionFermat's theorem on stationary points, statement, no evidence levelFermat's theorem on s…statementCauchy sequences are bounded, statement, no evidence levelCauchy sequences are …statementMonotone convergence theorem for sequences, statement, no evidence levelMonotone convergence …statementConvergent sequence, definition, no evidence levelConvergent sequencedefinitionEvery sequence has a monotone subsequence, statement, no evidence levelEvery sequence has a …statementPointwise convergence, definition, no evidence levelPointwise convergencedefinitionBolzano–Weierstrass theorem, statement, no evidence levelBolzano–Weierstrass t…statementUniform convergence, definition, no evidence levelUniform convergencedefinitionBoundedness theorem, statement, no evidence levelBoundedness theoremstatementUniform convergence on a set, definition, no evidence levelUniform convergence o…definitionCauchy criterion for sequences, statement, no evidence levelCauchy criterion for …statementExtreme value theorem, statement, no evidence levelExtreme value theoremstatementUniformly Cauchy sequence of functions, definition, no evidence levelUniformly Cauchy sequ…definitionRolle's theorem, statement, no evidence levelRolle's theoremstatementMean value theorem, statement, no evidence levelMean value theoremstatementUniform convergence of derivatives allows term-by-term differentiation, statement, no evidence levelUniform convergence o…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. Local maximum definition
  3. Sequence definition
  4. Limit of a function definition
  5. Local minimum definition
  6. Real sequence definition
  7. Subsequence definition
  8. Continuity at a point definition
  9. Derivative definition
  10. Bounded sequence definition
  11. Cauchy sequence definition
  12. Limit of a sequence definition
  13. Monotone sequence definition
  14. Continuous function definition
  15. Fermat's theorem on stationary points statement
  16. Cauchy sequences are bounded statement
  17. Monotone convergence theorem for sequences statement
  18. Convergent sequence definition
  19. Every sequence has a monotone subsequence statement
  20. Pointwise convergence definition
  21. Bolzano–Weierstrass theorem statement
  22. Uniform convergence definition
  23. Boundedness theorem statement
  24. Uniform convergence on a set definition
  25. Cauchy criterion for sequences statement
  26. Extreme value theorem statement
  27. Uniformly Cauchy sequence of functions definition
  28. Rolle's theorem statement
  29. Mean value theorem statement
  30. Uniform convergence of derivatives allows term-by-term differentiation statement

Where this leads

30 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Uniform_convergence, To differentiability
About this page
Tag0CS42KEM5C
URIm2://0CS42KEM5C
Kindstatement, theorem
WritingW0 stub
MSC40A30, 26A24
StatusThis copy of the site has no build data. Run m2 status 0CS42KEM5C.
UsesUniform convergence, Mean value theorem, Uniformly Cauchy sequence of functions

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