Extreme value theorem

A continuous real-valued function on a compact set attains both a maximum and a minimum. In Euclidean space, compactness is equivalent to being closed and bounded.

Compact space: A topological space in which every open cover has a finite subcover. Compactness supplies the key condition that keeps a function’s values from escaping their bounds.
Open cover: A collection of open sets whose union contains a given space or subset. Compactness is defined by requiring every open cover to have a finite subcover.
Extreme value theorem on a closed interval: The result that a continuous function on a closed bounded interval attains a maximum and minimum. This is the familiar real-variable form, obtained because closed bounded intervals are compact.
Weierstrass extreme value theorem for optimization: The use of compactness and continuity to guarantee a global optimizer exists. It ensures an optimization problem has an actual best solution before methods for finding it are applied.
Continuous function: A function for which sufficiently small changes in input produce arbitrarily small changes in output. Continuity ensures limiting function values are attained rather than lost at boundary points.
Closed set: A set whose complement is open, or equivalently one containing all its limit points in a metric space. Closedness is one part of the Euclidean characterization of compact sets.
Open interval: An interval that excludes one or both of its endpoints. A continuous function such as f(x)=x on (0,1) has no attained maximum or minimum.
Uniform continuity theorem: Every continuous real-valued function on a compact metric space is uniformly continuous. Compactness yields this companion result through a related subsequence argument.
Supremum: The least upper bound of a set of real numbers. The theorem turns the supremum of the function’s image into an attained maximum.
Bounded set: A set contained within some finite-radius ball in a metric space. Boundedness is the other part of that characterization in Euclidean space.

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