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Definition: A function f from the metric space (X, d_X) to the metric space (Y, d_Y) is said to be uniformly continuous if given epsilon > 0, there is delta > 0 such that for every pair of points x_0, x_1 of X, d_X(x_0, x_1) < delta implies d_Y(f(x_0), f(x_1)) < epsilon.
Source: Topology (second edition) by James R. Munkres