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A continuous real-valued function on a compact set has finite sup norm
Let be a metric space, let be compact, and let be continuous.
Proposition: The function is bounded on : there exists such that for all .
This proposition is one of the core reasons compactness is the right hypothesis for global control from local continuity.
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.
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
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 ,