Uniform Continuity

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.