Math 2.0

Real functions Theorem

Continuous functions on a closed bounded interval are uniformly continuous

Builds on Continuous functionUniform continuity on a setBolzano–Weierstrass theorem

Statement

Let a≤ba \le b and let f ⁣:[a,b]→ℝf \colon [a, b] \to \mathbb{R} be continuous. Then ff is uniformly continuous on [a,b][a, b].

Remarks

Suppose not. Then there is ε>0\varepsilon > 0 and points xn,ynx_n, y_n with ∣xn−yn∣<1/n|x_n - y_n| < 1/n and ∣f(xn)−f(yn)∣≥ε|f(x_n) - f(y_n)| \ge \varepsilon. A convergent subsequence of (xn)(x_n) leads to a contradiction with continuity. The interval must be closed and bounded. The function 1/x1/x is continuous on (0,1](0, 1] but not uniformly continuous there. The name honors Eduard Heine and Georg Cantor. Mathlib states it for compact sets, as IsCompact.uniformContinuousOn_of_continuous.

Reading path

To understand this page from the ground up, read these 18 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 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…statementConvergent 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 …statementUniform continuity, definition, no evidence levelUniform continuitydefinitionBolzano–Weierstrass theorem, statement, no evidence levelBolzano–Weierstrass t…statementUniform continuity on a set, definition, no evidence levelUniform continuity on…definitionContinuous functions on a closed bounded interval are uniformly continuous, statement, no evidence levelContinuous functions …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. Limit of a sequence definition
  9. Monotone sequence definition
  10. Continuous function definition
  11. Epsilon-delta definition of continuity statement
  12. Convergent sequence definition
  13. Monotone convergence theorem for sequences statement
  14. Every sequence has a monotone subsequence statement
  15. Uniform continuity definition
  16. Bolzano–Weierstrass theorem statement
  17. Uniform continuity on a set definition
  18. Continuous functions on a closed bounded interval are uniformly continuous statement

Where this leads

17 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Heine–Cantor_theorem, lead section
About this page
Tag0W0TZDRF9C
URIm2://0W0TZDRF9C
Kindstatement, theorem
WritingW0 stub
MSC26A15
StatusThis copy of the site has no build data. Run m2 status 0W0TZDRF9C.
UsesContinuous function, Uniform continuity on a set, Bolzano–Weierstrass theorem

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