Equicontinuity

Relation among continuous functions

In mathematical analysis, a family of functions is equicontinuous if all the functions are continuous and they have equal variation over a given neighbourhood, in a precise sense described herein. In particular, the concept applies to countable families, and thus sequences of functions.

Equicontinuity appears in the formulation of Ascoli's theorem, which states that a subset of C(X), the space of continuous functions on a compact Hausdorff space X, is compact if and only if it is closed, pointwise bounded and equicontinuous. As a corollary, a sequence in C(X) is uniformly convergent if and only if it is equicontinuous and converges pointwise to a function (not necessarily continuous a-priori). In particular, the limit of an equicontinuous pointwise convergent sequence of continuous functions f_n on either metric space or locally compact space^[1] is continuous. If, in addition, f_n are holomorphic, then the limit is also holomorphic.

The uniform boundedness principle states that a pointwise bounded family of continuous linear operators between Banach spaces is equicontinuous.^[2]

Equicontinuity between metric spaces

Let X and Y be two metric spaces, and F a family of functions from X to Y. We shall denote by d the respective metrics of these spaces.

The family F is equicontinuous at a point x₀ ∈ X if for every ε > 0, there exists a δ > 0 such that d(ƒ(x₀), ƒ(x)) < ε for all ƒ ∈ F and all x such that d(x₀, x) < δ. The family is pointwise equicontinuous if it is equicontinuous at each point of X.^[3]

The family F is uniformly equicontinuous if for every ε > 0, there exists a δ > 0 such that d(ƒ(x₁), ƒ(x₂)) < ε for all ƒ ∈ F and all x₁, x₂ ∈ X such that d(x₁, x₂) < δ.^[4]

For comparison, the statement 'all functions ƒ in F are continuous' means that for every ε > 0, every ƒ ∈ F, and every x₀ ∈ X, there exists a δ > 0 such that d(ƒ(x₀), ƒ(x)) < ε for all x ∈ X such that d(x₀, x) < δ.

• For continuity, δ may depend on ε, ƒ, and x₀.
• For uniform continuity, δ may depend on ε and ƒ.
• For pointwise equicontinuity, δ may depend on ε and x₀.
• For uniform equicontinuity, δ may depend only on ε.

More generally, when X is a topological space, a set F of functions from X to Y is said to be equicontinuous at x if for every ε > 0, x has a neighborhood U_x such that

${\displaystyle d_{Y}(f(y),f(x))<\epsilon }$

for all y ∈ U_x and ƒ ∈ F. This definition usually appears in the context of topological vector spaces.

When X is compact, a set is uniformly equicontinuous if and only if it is equicontinuous at every point, for essentially the same reason as that uniform continuity and continuity coincide on compact spaces. Used on its own, the term "equicontinuity" may refer to either the pointwise or uniform notion, depending on the context. On a compact space, these notions coincide.

Some basic properties follow immediately from the definition. Every finite set of continuous functions is equicontinuous. The closure of an equicontinuous set is again equicontinuous. Every member of a uniformly equicontinuous set of functions is uniformly continuous, and every finite set of uniformly continuous functions is uniformly equicontinuous.

Examples

• A set of functions with a common Lipschitz constant is (uniformly) equicontinuous. In particular, this is the case if the set consists of functions with derivatives bounded by the same constant.
• Uniform boundedness principle gives a sufficient condition for a set of continuous linear operators to be equicontinuous.
• A family of iterates of an analytic function is equicontinuous on the Fatou set.^[5][6]

Counterexamples

• The sequence of functions f_n(x) = arctan(nx), is not equicontinuous because the definition is violated at x₀=0.

Equicontinuity of maps valued in topological groups

Suppose that T is a topological space and Y is an additive topological group (i.e. a group endowed with a topology making its operations continuous). Topological vector spaces are prominent examples of topological groups and every topological group has an associated canonical uniformity.

Definition:^[7] A family H of maps from T into Y is said to be equicontinuous at t ∈ T if for every neighborhood V of 0 in Y, there exists some neighborhood U of t in T such that h(U) ⊆ h(t) + V for every h ∈ H. We say that H is equicontinuous if it is equicontinuous at every point of T.

Note that if H is equicontinuous at a point then every map in H is continuous at the point. Clearly, every finite set of continuous maps from T into Y is equicontinuous.

Equicontinuous linear maps

Because every topological vector space (TVS) is a topological group so the definition of an equicontinuous family of maps given for topological groups transfers to TVSs without change.

Characterization of equicontinuous linear maps

A family ${\displaystyle H}$ of maps of the form ${\displaystyle X\to Y}$ between two topological vector spaces is said to be equicontinuous at a point ${\displaystyle x\in X}$ if for every neighborhood ${\displaystyle V}$ of the origin in ${\displaystyle Y}$ there exists some neighborhood ${\displaystyle U}$ of the origin in ${\displaystyle X}$ such that ${\displaystyle h(x+U)\subseteq h(x)+V}$ for all ${\displaystyle h\in H.}$

If ${\displaystyle H}$ is a family of maps and ${\displaystyle U}$ is a set then let ${\displaystyle H(U):=\bigcup _{h\in H}h(U).}$ With notation, if ${\displaystyle U}$ and ${\displaystyle V}$ are sets then ${\displaystyle h(U)\subseteq V}$ for all ${\displaystyle h\in H}$ if and only if ${\displaystyle H(U)\subseteq V.}$

Let ${\displaystyle X}$ and ${\displaystyle Y}$ be topological vector spaces (TVSs) and ${\displaystyle H}$ be a family of linear operators from ${\displaystyle X}$ into ${\displaystyle Y.}$ Then the following are equivalent:

1. ${\displaystyle H}$ is equicontinuous;
2. ${\displaystyle H}$ is equicontinuous at every point of ${\displaystyle X.}$
3. ${\displaystyle H}$ is equicontinuous at some point of ${\displaystyle X.}$
4. ${\displaystyle H}$ is equicontinuous at the origin.
   • that is, for every neighborhood ${\displaystyle V}$ of the origin in ${\displaystyle Y,}$ there exists a neighborhood ${\displaystyle U}$ of the origin in ${\displaystyle X}$ such that ${\displaystyle H(U)\subseteq V}$ (or equivalently, ${\displaystyle h(U)\subseteq V}$for every ${\displaystyle h\in H}$).
   ^[8]
5. for every neighborhood ${\displaystyle V}$ of the origin in ${\displaystyle Y,}$