# 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 $\epsilon>0$ 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:

$\,$

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

 $\displaystyle d(f(x),f(x_{0}))<\epsilon$

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 $f_{n}(x):=\arctan(nx)$ 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. (ArzelÃÂ ’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.

## References

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