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Continuous functions on compact domains have finite upper and lower bounds
Corollary: Let be a compact metric space and let be continuous. Then is bounded: there exists such that for all .
Connection to parent theorem: By the extreme value theorem, attains a maximum and a minimum . Then .
A compact set is a subset of a topological space such that every open cover of contains a finite subcover: whenever is a family of open sets with , there exist indices such that
A metric space is a pair where is a set and is a metric on .
Every metric space determines a topological space via the metric-induced topology, whose basic neighborhoods are open balls.
Let be a function between metric spaces, and let . We say is continuous on if is continuous at every point (equivalently, the restriction is continuous).
Spelled out: for every and every , there exists such that for all ,
A bounded set in a metric space is a subset for which there exist and such that , where is the open ball of radius around .
The extreme value theorem states: if is a continuous function on a compact set , then attains its maximum and minimum values.
That is, there exist such that