Equicontinuity

an important property of some sets of functions. A set of functions is said to be equicontinuous on a given closed interval [a, b] if, for any number ∊ > 0, there exists a δ > 0 such that ǀf(x₂) - x₁ǀ < ∊ for any function f(x) of the given set whenever x_l and x₂ are in [a, b] and ǀx₂ – x_lǀ < δ. All functions of an equicontinuous set are uniformly continuous on [a, b].

The property of the equicontinuity of a set of functions finds application in the theory of differential equations and in functional analysis by virtue of the following theorem: for a uniformly convergent sequence of members of a given set of functions to exist, it is necessary and sufficient that the set of functions be equicontinuous and uniformly bounded—that is, that all the functions satisfy on [a, b) the condition ǀf(x)ǀ ≤ M with the same M. The possibility of singling out a uniformly convergent sequence means that the given set forms a relatively compact set in the space C of continuous functions.