# Equicontinuous functions

## Idea

A function $f$ is continuous if, roughly speaking, $f(x)$ is arbitrarily close to $f(y)$ whenever $x$ is sufficiently close to $y$. However, ‘close’ is relative, and $f(x)$ may be much closer to $f(y)$ than $g(x)$ is to $g(y)$, even if both $f$ and $g$ are continuous. Nevertheless, given a family of functions, we may have that $f(x)$ is arbitrarily close to $f(y)$ for every function $f$ in the family at once whenever $x$ is sufficiently close to $y$. In this case, the family of functions is equicontinuous.

## Definitions

Because we are considering the relative degree of closeness between potentially unrelated pairs of points, we need a uniform structure to define this concept. So let $X$ and $Y$ be uniform spaces (although the concept should make sense in somewhat greater generality), and let $\mathcal{F}$ be a family of functions from $X$ to $Y$.

###### Definition

The family $\mathcal{F}$ is continuous if each member is continuous: For each entourage $E$ in $Y$, for each function $f$ in $\mathcal{F}$ and each point $x \in X$, for some entourage $D$ in $X$, for each point $y$ in $X$, whenever $x \approx_D y$, we have $f(x) \approx_E f(y)$.

In short:

$\forall E,\; \forall f,\; \forall x,\; \exists D,\; \forall y,\; x \approx_D y \;\Rightarrow\; f(x) \approx_E f(y) .$
###### Definition

The family $\mathcal{F}$ is uniformly continuous if each member is uniformly continuous: For each entourage $E$ in $Y$, for each function $f$ in $\mathcal{F}$, for some entourage $D$ in $X$, for each point $x$ in $X$, for each point $y$ in $X$, whenever $x \approx_D y$, we have $f(x) \approx_E f(y)$.

In short:

$\forall E,\; \forall f,\; \exists D,\; \forall x,\; \forall y,\; x \approx_D y \;\Rightarrow\; f(x) \approx_E f(y) .$
###### Definition

The family $\mathcal{F}$ is equicontinuous if: For each entourage $E$ in $Y$, for each point $x$ in $X$, for some entourage $D$ in $X$, for each function $f$ in $\mathcal{F}$, for each point $y$ in $X$, whenever $x \approx_D y$, we have $f(x) \approx_E f(y)$.

In short:

$\forall E,\; \forall x,\; \exists D,\; \forall f,\; \forall y,\; x \approx_D y \;\Rightarrow\; f(x) \approx_E f(y) .$
###### Definition

The family $\mathcal{F}$ is uniformly equicontinuous if: For each entourage $E$ in $Y$, for some entourage $D$ in $X$, for each function $f$ in $\mathcal{F}$ and each point $x$ in $X$, for each point $y$ in $X$, whenever $x \approx_D y$, we have $f(x) \approx_E f(y)$.

In short:

$\forall E,\; \exists D,\; \forall f,\; \forall x,\; \forall y,\; x \approx_D y \;\Rightarrow\; f(x) \approx_E f(y) .$

All of these definitions are identical except for the placement of the quantifiers $\forall f$ and $\forall x$ before or after the quantifier $\exists D$.

Just as it is reasonable to speak of a single function $f$ continuous at a single point $x$, so it is reasonable to speak of a family equicontinuous at a single point $x$ (or, for that matter, of a single function $f$ uniformly continuous). However, it makes no sense to speak of a single function $f$ equicontinuous (or uniformly equicontinuous), nor can we speak of a family of functions uniformly equicontinuous at a single point $x$.

## Properties

### Cauchy sum theorem

The Cauchy sum theorem holds for an equicontinuous family of functions (from the real line to itself), without the requirement for uniform convergence.

### Arzela-Ascoli theorem

The importance of equicontinuity is perhaps best illustrated by the Arzelà-Ascoli theorem, which gives conditions for a set in a function space to be compact. A reasonably general version is this:

###### Theorem

Let $X$ be a convergence space and $Y$ a uniform space. Then a subset $F \subseteq C(X, Y)$ is relatively compact (has compact closure) iff it is equicontinuous and $\{f(x):\; f \in F\}$ is relatively compact in $Y$ for each $x \in X$.

See BB, corollary 2.4.9. Here the topology on the space $C(X, Y)$ of continuous functions $X \to Y$ is the so-called natural topology, namely the largest topology on $C(X, Y)$ such that for all spaces $A$, the continuity of a map $g: A \times X \to Y$ implies the continuity of its transpose $\hat{g}: A \to C(X, Y)$. This is the same as the exponential in $Top$ whenever the exponential exists. See Escardó, sections 8.1 and 10.2; see also ELS where it is shown that the natural topology on $C(X, Y)$ coincides with the topology of continuous convergence (which is the context for the theorem above).

(Some applications of Arzela-Ascoli should also be given.)

## References

• Wikipedia, Equicontinuity

• R. Beattie and H.-P. Butzmann, Convergence Structures and Applications to Functional Analysis, Kluwer Academic Publishers (2002).

• Martín Escardó, Synthetic topology of data types and classical spaces, Electronic Notes in Theor. Comp. Sci. (2004). (pdf)
• Martín Escardó, Jimmie Lawson, and Alex Simpson, Comparing cartesian closed categories of (core) compactly generated spaces, Topology and its Applications Vol. 143 Iss. 1-3 (2004), 105-145. (web)

Last revised on February 1, 2014 at 05:25:44. See the history of this page for a list of all contributions to it.