Uniform continuity

A function is uniformly continuous if, for every positive output tolerance, one positive input-distance bound works at every point in its domain.

Epsilon–delta definition: A formal definition of limits and continuity using positive tolerances and distance bounds. Uniform continuity places one input bound before the choice of domain point.
Sine function: The periodic function that assigns each real angle its sine value. Its bounded derivative makes it uniformly continuous on the real line.
Uniform convergence: Convergence of functions in which one index bound controls approximation at every point in the domain. Uniform limits of uniformly continuous functions remain uniformly continuous.
Metric space: A set equipped with a distance function satisfying nonnegativity, symmetry, and the triangle inequality. Uniform continuity is defined using distances in the domain and codomain.
Pointwise continuity: Continuity requiring a suitable input bound separately at each point of a function's domain. Unlike uniform continuity, its bound may shrink as the point changes.
Continuity: A function is continuous at a point when nearby inputs produce outputs close to its value there. Ordinary continuity allows the input bound to depend on the point.
Square-root function: The nonnegative square-root function mapping each nonnegative real number to its square root. It is uniformly continuous on its unbounded domain despite having an unbounded derivative near zero.
Completion of a metric space: A complete metric space containing a given metric space as a dense subspace. A uniformly continuous function extends uniquely to the completion when its codomain is complete.
Open cover: A collection of open sets whose union contains a specified space or subset. The Heine–Cantor theorem derives uniform continuity on compact spaces using finite subcovers.
Hölder continuity: A condition bounding output distances by a constant times a positive power of input distances. It gives a specific rate of uniform continuity when the exponent is positive.

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