Math 2.0

Sequences, series, summability Theorem

Uniform limit theorem

Builds on Uniform convergence on a setContinuity at a point

Statement

Let S⊆ℝS \subseteq \mathbb{R}, and let fn ⁣:S→ℝf_n \colon S \to \mathbb{R} be continuous for each nn. If fn→ff_n \to f uniformly on SS, then ff is continuous on SS.

Remarks

Fix c∈Sc \in S and ε>0\varepsilon > 0. Choose nn with ∣fn−f∣<ε/3|f_n - f| < \varepsilon/3 on SS, and δ\delta for the continuity of fnf_n at cc. Then ∣f(x)−f(c)∣≤∣f(x)−fn(x)∣+∣fn(x)−fn(c)∣+∣fn(c)−f(c)∣<ε|f(x) - f(c)| \le |f(x) - f_n(x)| + |f_n(x) - f_n(c)| + |f_n(c) - f(c)| < \varepsilon for ∣x−c∣<δ|x - c| < \delta. The [[sequence xnx^n on [0,1][0, 1]|0YSRR1JTKZ]] shows that pointwise convergence is not enough. Mathlib states the theorem as TendstoUniformly.continuous.

History

Augustin-Louis Cauchy claimed in 1821 that a series of continuous functions has a continuous sum, whenever it converges. In 1826, Niels Henrik Abel found counterexamples among Fourier series. In modern terms, Cauchy's argument needs uniform convergence.

Reading path

To understand this page from the ground up, read these 10 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 sequencedefinitionContinuity at a point, definition, no evidence levelContinuity at a pointdefinitionLimit of a sequence, definition, no evidence levelLimit of a sequencedefinitionPointwise convergence, definition, no evidence levelPointwise convergencedefinitionUniform convergence, definition, no evidence levelUniform convergencedefinitionUniform convergence on a set, definition, no evidence levelUniform convergence o…definitionUniform limit theorem, statement, no evidence levelUniform limit theoremstatement
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.
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  1. Limit point of a set definition
  2. Sequence definition
  3. Limit of a function definition
  4. Real sequence definition
  5. Continuity at a point definition
  6. Limit of a sequence definition
  7. Pointwise convergence definition
  8. Uniform convergence definition
  9. Uniform convergence on a set definition
  10. Uniform limit theorem statement

Where this leads

6 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Uniform_limit_theorem, Statement
About this page
Tag0X9XT05JVC
URIm2://0X9XT05JVC
Aliasesuniform-limit-of-continuous-functions-is-continuous
Kindstatement, theorem
WritingW1 stated
MSC40A30, 26A15
StatusThis copy of the site has no build data. Run m2 status 0X9XT05JVC.
UsesUniform convergence on a set, Continuity at a point
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