Math 2.0

Real functions Theorem

Extreme value theorem

Builds on Boundedness theoremContinuous function

Statement

Let a≤ba \le b and let f ⁣:[a,b]→ℝf \colon [a, b] \to \mathbb{R} be continuous. Then there are c,d∈[a,b]c, d \in [a, b] with f(c)≤f(x)≤f(d)f(c) \le f(x) \le f(d) for all x∈[a,b]x \in [a, b].

Remarks

By the boundedness theorem, M=sup⁡f([a,b])M = \sup f([a, b]) is finite. If f(x)<Mf(x) < M for all xx, then 1/(M−f(x))1/(M - f(x)) is continuous and unbounded, which is a contradiction. Each hypothesis is needed. The function xx on [0,1)[0, 1) has no maximum, and xx on [0,∞)[0, \infty) is not bounded above. Mathlib states it for compact sets, as IsCompact.exists_isMinOn and IsCompact.exists_isMaxOn.

History

Bernard Bolzano proved the theorem in the 1830s. His proof stayed unpublished until 1930.

Reading path

To understand this page from the ground up, read these 16 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 functiondefinitionConvergent 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 …statementBolzano–Weierstrass theorem, statement, no evidence levelBolzano–Weierstrass t…statementBoundedness theorem, statement, no evidence levelBoundedness theoremstatementExtreme value theorem, statement, no evidence levelExtreme value 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.
  • 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. Convergent sequence definition
  12. Monotone convergence theorem for sequences statement
  13. Every sequence has a monotone subsequence statement
  14. Bolzano–Weierstrass theorem statement
  15. Boundedness theorem statement
  16. Extreme value theorem statement

Where this leads

121 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Extreme_value_theorem, lead section
About this page
Tag0WS1A8E2P9
URIm2://0WS1A8E2P9
Kindstatement, theorem
WritingW1 stated
MSC26A15
StatusThis copy of the site has no build data. Run m2 status 0WS1A8E2P9.
UsesBoundedness theorem, Continuous function
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