Equicontinuity

In mathematical analysis, a sequence of functions is equicontinuous if all the functions are continuous and they have equal variation over a given neighbourhood (a precise definition appears below).

If a sequence of continuous functions converges pointwise, then the limit is not necessarily continuous (a counterexample is given by the family defined by f_n(x) = arctan nx, which converges to the discontinuous sign function). However, if the sequence is equicontinuous, then we can conclude that the limit is continuous.

Definitions

Let {f_n} be a sequence of functions from X ⊂ R to R (more general functions are considered below).

The sequence {f_n} is equicontinuous if for every ε > 0 and every x ∈ X, there exists a δ > 0, such that for all n and all x′ ∈ X with |x′ − x| < δ we have |f_n(x) − f_n(x′)| < ε.

The sequence {f_n} is uniformly equicontinuous if for every ε > 0, there exists a δ > 0, such that for all n and all x,x′ ∈ X with |x′ − x| < δ we have |f_n(x) − f_n(x′)| < ε.

For comparison, the statement all functions f_n are continuous means that for every ε > 0, every n, and every x ∈ X, there exists a δ > 0, such that for all x′ ∈ X with |x′ − x| < δ we have |f_n(x) − f_n(x′)| < ε. So, for continuity, δ may depend on ε, x and n; for equicontinuity, δ must be independent of n; and for uniform equicontinuity, δ must be independent of both n and x.

Properties

As promised in the introduction, the limit of a pointwise convergent, equicontinuous sequence is continuous.

Theorem 1: Let {f_n} be an equicontinuous sequence of functions. If f_n(x) → f(x) for every x ∈ X, then the function f is continuous.

The condition in the above theorem can be slightly weakened. It suffices if the sequence converges pointwise on a dense subset.

Theorem 2: Let {f_n} be an equicontinuous sequence of functions from X ⊂ R to R. Suppose that f_n(x) converges for all x ∈ D, where D is a dense subset of X. Then, f_n(x) converges for all x ∈ X, and the limit function is continuous.

If the domain of the functions f_n is the closed interval [0, 1], we can say a bit more. Firstly, the properties of equicontinuity and uniform equicontinuity are equivalent.

Theorem 3: Every equicontinuous sequence of functions from [0, 1] to R is uniformly equicontinuous.

Furthermore, equicontinuity and pointwise convergence imply uniform convergence.

Theorem 4: Let {f_n} be an equicontinuous sequence of functions from [0, 1] to R. If f_n(x) → f(x) for every x ∈ [0, 1], then f_n(x) → f(x) uniformly in x.

The final result can be viewed as a generalization of the Bolzano-Weierstrass theorem to functions.

Ascoli's theorem: Let {f_n} be an equicontinuous sequence of uniformly bounded functions from [0, 1] to R. Then there is a subsequence which converges uniformly.

The term uniformly bounded means that |f_n(x)| < C for some C, independent of x and n.

Generalizations

The definition for equicontinuity generalizes to functions between arbitrary metric spaces. Suppose that {f_n} is a sequence of functions from X to Y. This sequence is equicontinuous if for every ε > 0 and every x ∈ X, there exists a δ, such that for all n and all x′ ∈ X with d_X(x, x′) < δ we have d_Y(f_n(x), f_n(x′)) < ε, where d_X and d_Y denote the metrics on X and Y, respectively. The definition for uniform equicontinuity can be generalized in the same manner.

Theorem 1 is still valid in this setting, but Theorem 2 only holds if the codomain Y is complete.

The most general scenario in which equicontinuity can be defined is for topological spaces whereas uniform equicontinuity requires the filter of neighbourhoods of one point to be somehow comparable with the filter of neighbourhood of another point. The latter is most generally done via a uniform structure, giving a uniform space. Appropriate definitions in these cases are as follows:

A set of functions A between two topological spaces X and Y is equicontinuous at the point x ∈ X if for every neighbourhood V of f(x), there is some neighbourhood U of x such that for every function f in A, f(U) ⊆ V. (Equivalently, the intersection of all f^-1(V) is a neighbourhood of x).

A set of functions A between two topological spaces X and Y is equicontinuous if it is equicontinuous at x, for all points x ∈ X.

A set of functions A between two uniform spaces X and Y is uniformly equicontinuous if for every element W of the uniformity on Y, the set

{ (u,v) ∈ X × X: for all f ∈ A. (f(u),f(v)) ∈ W }

is a member of the uniformity on X

For metric spaces, there are standard topologies and uniform structures derived from the metrics, and then these general definitions are equivalent to the metric-space definitions.