Equicontinuity

Equicontinuity is a property of a family of functions that expresses a uniform version of continuity across the whole family.

Let (X, dX) and (Y, dY) be metric spaces and F be a family of functions from X to Y. F is equicontinuous at a point x0 ∈ X if, for every ε > 0, there exists δ > 0 such that for all f ∈ F and all x ∈ X with dX(x, x0) < δ we have dY(f(x), f(x0)) < ε. In words, the same δ works for all functions in F at the given base point x0.

F is equicontinuous on a subset E ⊆ X if the above holds for every x0 ∈ E. F is uniformly equicontinuous on E if, for every ε > 0, there exists δ > 0 such that for all f ∈ F and all x, y ∈ E with dX(x, y) < δ we have dY(f(x), f(y)) < ε. Uniform equicontinuity is a stronger condition than equicontinuity at each point, because δ is chosen independently of the base point and of the particular pair (x, y).

Examples and notes: If F = {f_t(x) = t x} with t taken from a bounded set of real numbers and X = Y = R, then F is equicontinuous on R; if t is unbounded, equicontinuity may fail. Translations f_a(x) = x + a form a uniformly equicontinuous family on R, since |f_a(x) - f_a(y)| = |x - y|. Equicontinuity plays a central role in compactness results for function spaces, notably the Arzelà–Ascoli theorem: on a compact X, a family F ⊂ C(X, R) that is equicontinuous and pointwise bounded has compact closure in the sup norm. The concept extends to other target spaces and to uniform spaces, where it describes uniform control of oscillation across the family.

In short, equicontinuity captures the idea that a family of functions cannot oscillate more wildly than a single rate of change, uniformly across its members.