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 a metric space or a locally compact space 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.

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.

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.

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.

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