Heine–Cantor theorem

Every continuous function from a compact metric space to a metric space is uniformly continuous. Compactness makes the continuity bounds work uniformly across the entire domain.

Uniform continuity: A function is uniformly continuous when one input-distance bound controls output variation throughout its domain. This is the conclusion the theorem guarantees on compact metric domains.
Continuity: A function is continuous when sufficiently small input changes produce small output changes locally at every point. Continuity supplies the local control required by the theorem.
Continuous function on a noncompact domain: A continuous function whose domain lacks compactness, so uniform continuity need not follow. It shows that continuity alone does not guarantee the theorem’s conclusion.
Extreme value theorem: A theorem stating that a continuous real-valued function on a compact space attains its maximum and minimum. It is a companion consequence of compactness and continuity, with a different global guarantee.
Eduard Heine: A German mathematician whose work included foundational results in analysis. His name is attached to the theorem’s compactness-to-uniformity result.
Compactness: A topological property that, in metric spaces, is equivalent to every open cover having a finite subcover. Finite subcovers let local continuity neighborhoods yield a global uniform bound.
Metric space: A set equipped with a distance function satisfying nonnegativity, identity, symmetry, and the triangle inequality. Metrics define the distances used in both continuity and uniform continuity.
x squared: The real function mapping each number x to its square. It is continuous on the real line but not uniformly continuous there.
Continuous image of a compact space: The image of a compact space under a continuous map is compact. Together with the theorem, it supplies global control over both input and output spaces.
Georg Cantor: A German mathematician who founded set theory and made major contributions to real analysis. His name is attached to the theorem and its compactness framework.

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