equicontinuous
1 Definition
Let $X$ be a topological space^{}, $(Y,d)$ a metric space and $C(X,Y)$ the set of continuous functions^{} $X\to Y$.
Let $\mathcal{F}$ be a subset of $C(X,Y)$. A function $f\in \mathcal{F}$ is continuous at a point ${x}_{0}$ when given $\u03f5>0$ there is a neighbourhood $U$ of ${x}_{0}$ such that $$ for every $x\in U$. When the same neighbourhood $U$ can be chosen for all functions $f\in \mathcal{F}$, the family $\mathcal{F}$ is said to be equicontinuous. More precisely:
$$
Definition  Let $\mathcal{F}$ be a subset of $C(X,Y)$. The set of functions $\mathcal{F}$ is said to be equicontinuous at ${x}_{0}\in X$ if for every $\u03f5>0$ there is a neighbourhood $U$ of ${x}_{0}$ such that for every $x\in U$ and every $f\in \mathcal{F}$ we have
$$ 
The set $\mathcal{F}$ is said to be equicontinuous if it is equicontinuous at every point $x\in X$.
2 Examples

•
A finite set^{} 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 ${\{{f}_{n}\}}_{n\in \mathbb{N}}$, where ${f}_{n}:\mathbb{R}\to \mathbb{R}$ is given by ${f}_{n}(x):=\mathrm{arctan}(nx)$ is not equicontinuous at $0$.
3 Properties

•
If a subset $\mathcal{F}\subseteq C(X,Y)$ is totally bounded^{} under the uniform metric, then $\mathcal{F}$ is equicontinuous.

•
Suppose $X$ is compact^{}. If a sequence of functions $\{{f}_{n}\}$ in $C(X,{\mathbb{R}}^{k})$ is equibounded and equicontinuous, then the sequence $\{{f}_{n}\}$ has a uniformly convergent subsequence. (ArzelÃÂ ’s theorem (http://planetmath.org/AscoliArzelaTheorem))

•
Let $\{{f}_{n}\}$ be a sequence of functions in $C(X,Y)$. If $\{{f}_{n}\}$ is equicontinuous and converges^{} pointwise^{} to a function $f:X\to Y$, then $f$ is continuous and $\{{f}_{n}\}$ converges to $f$ in the compactopen topology^{}.
References
 1 J. Munkres, Topology (2nd edition), Prentice Hall, 1999.
Title  equicontinuous 

Canonical name  Equicontinuous 
Date of creation  20130322 18:38:10 
Last modified on  20130322 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 