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 $\epsilon > 0$ there is a neighbourhood $U$ of $x_0$ such that $d(f(x),f(x_0)) < \epsilon$ 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:

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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 $\epsilon >0$ there is a neighbourhood $U$ of $x_0$ such that for every $x \in U$ and every $f \in \mathcal{F}$ we have \begin{align*} d(f(x),f(x_0)) < \epsilon \end{align*} The set $\mathcal{F}$ is said to be equicontinuous if it is equicontinuous at every point $x \in X$ .

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 $f_n(x):=\arctan (nx)$ is not equicontinuous at $0$ .

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. (Arzelà's theorem)
• 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.

Bibliography

1

J. Munkres, Topology (2nd edition), Prentice Hall, 1999.