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Definition: Let (Y, d) be a metric space. Let F be a subset of the function space C(X, Y). If x_0 is in X, the set F of functions is said to be equicontinuous at x_0 if given epsilon greater than 0, there is a neighborhood U of x_0 such that for all x in U and all f in F, d(f(x), f(x_0)) is less than epsilon. If the set F is equicontinuous at x_0 for each x_0 in X, it is said simply to be equicontinuous.
Source: Topology (second edition) by James R. Munkres