Equicontinuity
In mathematical analysis, a sequence of functions is equicontinuous if all the functions are continuous and they are all, in some sense, "equally" convergent (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 fn(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 {fn} be a sequence of functions from X ⊂ R to R (more general functions are considered below).
The sequence {fn} 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 |fn(x) − fn(x′)| < ε.
The sequence {fn} 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 |fn(x) − fn(x′)| < ε.
For comparison, the statement all functions fn 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 |fn(x) − fn(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 {fn} be an equicontinuous sequence of functions. If fn(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 {fn} be an equicontinuous sequence of functions from X ⊂ R to R. Suppose that fn(x) converges for all x ∈ D, where D is a dense subset of X. Then, fn(x) converges for all x ∈ X, and the limit function is continuous.
If the domain of the functions fn 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 {fn} be an equicontinuous sequence of functions from [0, 1] to R. If fn(x) → f(x) for every x ∈ [0, 1], then fn(x) → f(x) uniformly in x.
The final result can be viewed as a generalization of the theorem of Bolzano-Weierstrass to functions.
Ascoli's theorem: Let {fn} 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 |fn(x)| < C for some C, independent of x and n.
Generalization
The definition for equicontinuity generalizes to functions between arbitrary metric spaces. Suppose that {fn} 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 dX(x, x′) < δ we have dY(fn(x), fn(x′) < ε, where dX and dY 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.
Referenced By
List of mathematical topics (D-F) | List of mathematical topics (F-Z)
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