Math 2.0

Sequences, series, summability Corollary

Continuity of the sum of a uniformly convergent series of functions

Builds on Series of functionsUniform limit theoremWeierstrass M-test

Statement

Let S⊆ℝS \subseteq \mathbb{R}, and let fn ⁣:S→ℝf_n \colon S \to \mathbb{R} be continuous for each n≥0n \ge 0. If the series ∑n=0∞fn\sum_{n=0}^{\infty} f_n converges uniformly on SS, then its sum is continuous on SS.

Remarks

The partial sums are continuous, and they converge uniformly. Apply the uniform limit theorem. Uniform convergence often comes from the M-test. Mathlib states the result in this form as continuous_tsum. For example, ∑n=1∞cos⁡(nx)n2\sum_{n=1}^{\infty} \frac{\cos(n x)}{n^2} is continuous on ℝ\mathbb{R}.

Reading path

To understand this page from the ground up, read these 29 pages in order. The foundations come first.

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Limit point of a set, definition, no evidence levelLimit point of a setdefinitionSequence, definition, no evidence levelSequencedefinitionLimit of a function, definition, no evidence levelLimit of a functiondefinitionReal sequence, definition, no evidence levelReal sequencedefinitionSubsequence, definition, no evidence levelSubsequencedefinitionContinuity at a point, definition, no evidence levelContinuity at a pointdefinitionBounded 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 sequencedefinitionSeries, definition, no evidence levelSeriesdefinitionCauchy 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 convergencedefinitionPartial sum, definition, no evidence levelPartial sumdefinitionBolzano–Weierstrass theorem, statement, no evidence levelBolzano–Weierstrass t…statementConvergent series, definition, no evidence levelConvergent seriesdefinitionUniform convergence, definition, no evidence levelUniform convergencedefinitionCauchy criterion for sequences, statement, no evidence levelCauchy criterion for …statementUniform convergence on a set, definition, no evidence levelUniform convergence o…definitionA series of nonnegative terms converges iff its partial sums are bounded, statement, no evidence levelA series of nonnegati…statementSeries of functions, definition, no evidence levelSeries of functionsdefinitionUniform limit theorem, statement, no evidence levelUniform limit theoremstatementUniformly Cauchy sequence of functions, definition, no evidence levelUniformly Cauchy sequ…definitionComparison test, statement, no evidence levelComparison teststatementWeierstrass M-test, statement, no evidence levelWeierstrass M-teststatementContinuity of the sum of a uniformly convergent series of functions, statement, no evidence levelContinuity of the sum…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. Sequence definition
  3. Limit of a function definition
  4. Real sequence definition
  5. Subsequence definition
  6. Continuity at a point definition
  7. Bounded sequence definition
  8. Cauchy sequence definition
  9. Limit of a sequence definition
  10. Monotone sequence definition
  11. Series definition
  12. Cauchy sequences are bounded statement
  13. Monotone convergence theorem for sequences statement
  14. Convergent sequence definition
  15. Every sequence has a monotone subsequence statement
  16. Pointwise convergence definition
  17. Partial sum definition
  18. Bolzano–Weierstrass theorem statement
  19. Convergent series definition
  20. Uniform convergence definition
  21. Cauchy criterion for sequences statement
  22. Uniform convergence on a set definition
  23. A series of nonnegative terms converges iff its partial sums are bounded statement
  24. Series of functions definition
  25. Uniform limit theorem statement
  26. Uniformly Cauchy sequence of functions definition
  27. Comparison test statement
  28. Weierstrass M-test statement
  29. Continuity of the sum of a uniformly convergent series of functions statement

Where this leads

4 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Weierstrass_M-test, Statement
About this page
Tag0R0MYT0BJD
URIm2://0R0MYT0BJD
Kindstatement, corollary
WritingW0 stub
MSC40A30, 26A15
StatusThis copy of the site has no build data. Run m2 status 0R0MYT0BJD.
UsesSeries of functions, Uniform limit theorem, Weierstrass M-test

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