Uniform Continuity

(redirected from Uniformly continuous)

uniform continuity

[′yü·nə‚fȯrm känt·ən′ü·əd·ē]
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.

Mentioned in ?
References in periodicals archive ?
Then it is uniformly continuous and bounded on T x D; here D is any compact subset of X.
nm](t) is absolutely uniformly continuous on I where m, n [member of] N [union] {0}.
The topics include the strongly bounded Turing degrees of simple sets, recursive aspects of diophantine properties of Brownian motion, complexity issues for pre-orders on finite labeled forests, Lipschitz and uniformly continuous reducibilities on ultrametric Polish spaces, the equivalence of paraconsistent and explosive versions of Nelson logic, and an isomorphism theorem for partial numberings.
Let the phase space B = C((-[infinity], 0], E), the space of bounded uniformly continuous functions endowed with the following norm:
Mogbademu, Modified Noor iterative procedure for uniformly continuous mappings in Banach spaces, Boletin de la Asociacion Matematica Venezolana, XVIII(2011), No.
In this paper, our purpose is to show that the more general modified Mann iteration sequence with errors converges to the unique fixed point of T if T : X [right arrow] X is a uniformly continuous strongly successively pseudocontractive mapping with a bounded range or T : X [right arrow] X is uniformly Lipschitzian and strongly successively pseudocontractive mapping without necessarily having a bounded range.
the functions [gamma](t), g(t) are uniformly continuous and 0 [less than or equal to] [gamma](t) [less than or equal to] g(t) [less than or equal to]t;
Fibrewise uniform spaces were studied by Niefield where both base space and the total space are required to be uniform and the projection is required to be uniformly continuous.
It is well-known that a Banach space E is uniformly smooth if and only if the duality map J is single-valued and norm-to-norm uniformly continuous on bounded sets of E.
Juergen Pohlenz, engineering manager at ZF Passau in Germany, explained that CVTs on such vehicles operating with heavy loads have the advantage of a uniformly continuous increase of traction force from a standstill, which is particularly welcomed when operating on difficult grounds.
Since f is uniformly continuous on compact subsets of [0, 1] x (0, c/[delta]].
n] (s) is uniformly continuous on the interval [0,1 - [gamma]], we derive that each term on the right hand side of (2.

Full browser ?