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uniform continuity[′yü·nə‚fȯrm känt·ən′ü·əd·ē]
an important concept in mathematical analysis. A function f(x) is said to be uniformly continuous on a given set if for every ∊ > 0, it is possible to find a number δ = δ(∊) > 0 such that ǀf(x1) - f(x2)ǀ < ∊ for any pair of numbers x1 and x2 of the given set satisfying the condition ǀx1 - x2ǀ < δ (see). For example, the function f(x) = x2, is uniformly continuous on the closed interval [0, 1] because if ǀx1 - x2ǀ < ∊/2, then ǀf(x1) – f(x2)ǀ = ǀx1 - x2 ǀǀ x1 + x2 ǀ < ∊ (since necessarily ǀx1 + x2ǀ ≤ 2 when 0 ≤ x1 ≤ 1 and 0 ≤ x2 ≤ 1).
According to Cantor’s theorem, a function continuous at every point of a closed interval [a, b] is in general uniformly continuous on the interval. This theorem may not hold for an open interval. For example, the function f(x) = 1/x is continuous at every point of the interval 0 < x < 1 but is not uniformly continuous on the interval. This can be shown as follows. Suppose ∊ = 1. For any δ > 0 (δ < 1), the numbers x1 = δ/2 and x2 = δ satisfy the inequality ǀx1 - x2ǀ < δ, but ǀf(x1)- f(x2)ǀ = 1/δ > 1.