1 Definition

Let X be a topological spaceMathworldPlanetmath, (Y,d) a metric space and C(X,Y) the set of continuous functionsMathworldPlanetmathPlanetmath XY.

Let be a subset of C(X,Y). A function f is continuous at a point x0 when given ϵ>0 there is a neighbourhood U of x0 such that d(f(x),f(x0))<ϵ for every xU. When the same neighbourhood U can be chosen for all functions f, the family is said to be equicontinuous. More precisely:

Definition - Let be a subset of C(X,Y). The set of functions is said to be equicontinuous at x0X if for every ϵ>0 there is a neighbourhood U of x0 such that for every xU and every f we have


The set is said to be equicontinuous if it is equicontinuous at every point xX.

2 Examples

  • A finite setMathworldPlanetmath of functions in C(X,Y) is always equicontinuous.

  • When X is also a metric space, a family of functions in C(X,Y) that share the same Lipschitz constant is equicontinuous.

  • The family of functions {fn}n, where fn: is given by fn(x):=arctan(nx) is not equicontinuous at 0.

3 Properties


  • 1 J. Munkres, Topology (2nd edition), Prentice Hall, 1999.
Title equicontinuous
Canonical name Equicontinuous
Date of creation 2013-03-22 18:38:10
Last modified on 2013-03-22 18:38:10
Owner asteroid (17536)
Last modified by asteroid (17536)
Numerical id 6
Author asteroid (17536)
Entry type Definition
Classification msc 54E35
Classification msc 54C35
Synonym equicontinuity
Related topic Equicontinuous2