Uniform Continuity

(redirected from Uniformly continuous function)

uniform continuity

[′yü·nə‚fȯrm känt·ən′ü·əd·ē]
(mathematics)
A property of a function ƒ on a set, namely: given any ε > 0 there is a δ > 0 such that |ƒ(x1) - ƒ(x2)| < ε="" provided="" |="">x1-x2| < δ="" for="" any="" pair="">x1, x2 in the set.

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.

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Now, using the constructive version of KKM lemma proved in the previous section, we shall show that there exists an approximate fixed point for any uniformly continuous function from an n-dimensional simplex to itself.
A right uniformly continuous is defined similarly; by a uniformly continuous function, we shall mean a left and right uniformly continuous function.
Let the phase space B = C((-[infinity], 0], E), the space of bounded uniformly continuous functions endowed with the following norm:

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