Editing Equicontinuity (section)

{{Short description|Relation among continuous functions}}
In [[mathematical analysis]], a family of functions is '''equicontinuous''' if all the functions are [[continuous function|continuous]] and they have equal [[Bounded variation|variation]] over a given [[neighbourhood (mathematics)|neighbourhood]], in a precise sense described herein. 
In particular, the concept applies to [[countable set|countable]] families, and thus ''sequences'' of functions.

Equicontinuity appears in the formulation of [[Ascoli's theorem]], which states that a subset of ''C''(''X''), the space of [[continuous functions on a compact Hausdorff space]] ''X'', is compact if and only if it is closed, pointwise bounded and equicontinuous. 
As a corollary, a sequence in ''C''(''X'') is uniformly convergent if and only if it is equicontinuous and converges pointwise to a function (not necessarily continuous a-priori). 
In particular, the limit of an equicontinuous pointwise convergent sequence of continuous functions ''f<sub>n</sub>'' on either a metric space or a [[locally compact space]]<ref>More generally, on any [[compactly generated space]]; e.g., a [[first-countable space]].</ref> is continuous. If, in addition, ''f<sub>n</sub>'' are [[holomorphic]], then the limit is also holomorphic.

The uniform boundedness principle states that a pointwise bounded family of continuous linear operators between Banach spaces is equicontinuous.{{sfn|Rudin|1991|p=44 §2.5}}