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 $ℱ$ be a subset of $C(X,Y)$. A function $f\in ℱ$ is continuous at a point $x_0$ when given $ϵ>0$ 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 $f\in ℱ$, the family $ℱ$ is said to be equicontinuous. More precisely:

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Definition - Let $ℱ$ be a subset of $C(X,Y)$. The set of functions $ℱ$ is said to be equicontinuous at $x_0\in X$ if for every $ϵ>0$ there is a neighbourhood $U$ of $x_0$ such that for every $x\in U$ and every $f\in ℱ$ we have

The set $ℱ$ 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 $f_n:ℝ\to ℝ$ is given by $f_n(x):=\arctan (nx)$ is not equicontinuous at $0$.

3 Properties

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

• Suppose $X$ is compact. If a sequence of functions $\{f_n\}$ in $C(X,{ℝ}^k)$ is equibounded and equicontinuous, then the sequence $\{f_n\}$ has a uniformly convergent subsequence. (Arzelà’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 compact-open topology.