# Equicontinuity

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## 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(*x*_{2}) - *x*_{1}ǀ < ∊ for any function *f(x*) of the given set whenever *x*_{l} and *x*_{2} are in [*a, b*] and ǀ*x*_{2} – *x*_{l}ǀ < 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 ǀ*f*(*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.