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The 22 pages that link to Extreme value theorem, each with the reason it gives.

• Continuous functionRelated: It shows how continuity and compactness guarantee actual extrema.

• Compact spaceRelated: Compactness ensures continuous functions cannot escape their extrema.

• ContinuityRelated: Continuity guarantees extrema when the domain is compact.

• SupremumRelated: Compactness converts a finite supremum into an attained maximum.

• Intermediate value theoremRelated: It shares continuity as a hypothesis but guarantees extrema rather than all intermediate values.

• Bolzano–Weierstrass theoremRelated: The theorem helps prove extrema exist by extracting limits from optimizing sequences.

• Heine–Borel theoremRelated: The theorem lets closed bounded Euclidean domains meet the compactness hypothesis for extrema.

• Bounded functionRelated: Compactness and continuity guarantee boundedness and actual attainment of extreme values.

• Bounded setRelated: In Euclidean space, closed bounded sets are compact, enabling this conclusion.

• Mean value theoremRelated: Its compactness guarantee is part of a standard proof of the mean value theorem.

• Rolle's theoremRelated: It ensures the function reaches extrema that can force a stationary point.

• Sequential compactnessRelated: Sequential compactness supplies the familiar sequence-based proof in metric settings.

• Darboux's theoremRelated: Extrema on a closed interval help establish that derivatives cannot skip intermediate values.

• Upper boundRelated: Compactness turns bounded ranges into attained extrema under continuity.

• Completeness of the real numbersRelated: Completeness helps ensure that bounded continuous images contain their endpoint values.

• Maximum and minimumRelated: Its hypotheses give a central condition ensuring global extrema exist.

• Heine–Cantor theoremRelated: It is a companion consequence of compactness and continuity, with a different global guarantee.

• Bolzano's theoremRelated: It is another consequence of continuity on a closed interval, but concerns extrema rather than sign changes.