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What is "uniformly continuous"

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uniformly continuous

a. (context analysis of a function from a metric space X to a metric space Y English) That for every real number ''ε'' > 0 there exists a real ''δ'' > 0 such that for all pairs of points ''x'' and ''y'' in ''X'' for which D_X (x, y) < delta , it must be the case that D_Y (f(x), f(y)) < epsilon (where ''D''''X'' and ''D''''Y'' are the metrics of ''X'' and ''Y'', respectively).