Uniform Continuity

(redirected from Uniform 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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If f is both (r, s) intuitionistic Fuzzy quasi uniform increasing continuous function and (r, s) intuitionistic Fuzzy quasi uniform decreasing continuous function then it is called ordered (r, s) intuitionistic Fuzzy quasi uniform continuous function.
Then f is said to be lower (r, s) intuitionistic Fuzzy quasi uniform continuous function if [f.
Then f is said to be upper (r, s) intuitionistic Fuzzy quasi uniform continuous function if [f.
Then f is lower (upper) (r, s) intuitionistic Fuzzy quasi uniform continuous function if and only if A is an intuitionistic Fuzzy smooth quasi uniform increasing (decreasing) open [F.
I](I), g is an lower (r, s) intuitionistic Fuzzy quasi uniform continuous function, h is an upper (r, s) intuitionistic Fuzzy quasi uniform continuous function and g [subset or equal to] h, then there exists an (r, s) intuitionistic Fuzzy quasi uniform continuous function f: (X, [T.
A] lower and upper (r, s) intuitionistic Fuzzy quasi uniform continuous function respectively.

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