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Uniform Continuity
Published in John D. Ross, Kendall C. Richards, Introductory Analysis, 2020
John D. Ross, Kendall C. Richards
Based on your work from the exercise above, it should be clear that a continuous function does not always preserve the Cauchy property of a given sequence in its domain. ⇝ In this chapter, we will introduce a stronger version of continuity, called uniform continuity, which attempts to address these limitations. Intuitively, uniform continuity is continuity in which the relationship between ε and δ is the same throughout the entire domain (in other words, the ε - δ relationship is uniform). We will see that any function that satisfies this new definition will automatically satisfy our old definition (i.e., if a function is uniformly continuous on a set D, then it is continuous on D “in the original way”), but that the converse of this statement is not true.
Limits and Continuity
Published in Hemen Dutta, Pinnangudi N. Natarajan, Yeol Je Cho, Concise Introduction to Basic Real Analysis, 2019
Hemen Dutta, Pinnangudi N. Natarajan, Yeol Je Cho
The notion of continuity of a function is local in nature in the sense that it is defined for individual points of the domain. Even when we check continuity on the whole domain, we do it by checking continuity at every point of the domain individually. In contrast, the notion of uniform continuity of function is global in nature, in the sense that it is defined for pairs of points of the domain rather than individual points and thus applicable over the whole domain.
Preliminaries
Published in Ronald B. Guenther, John W. Lee, Sturm-Liouville Problems, 2018
Ronald B. Guenther, John W. Lee
A real or complex-valued function f defined on a set S in Euclidean space is continuous at x0 in S if given any ε>0 there is a δ > 0, dependent on x0 and on ε, such that |f(x)−f(x0)|<ε whenever x ∈ S satisfies |x−x0|<δ. A function f is continuous on FunctioncontinuousContinuityS if it is continuous at every point of S. A function f is uniformly continuous onFunctionuniformly continuousUniform continuityS if given any ε>0 there is a δ > 0, dependent only on ε, such that |f(x)−f(x′)|<ε whenever x, x′∈S satisfy |x−x′|<δ. Uniform continuity on a set S means that there is a single δ > 0 in the definition of continuity of f at x0 that works simultaneously for all x0 in S.
Asymptotic behaviour of a nonautonomous evolution equation governed by a quasi-nonexpansive operator
Published in Optimization, 2022
Ming Zhu, Rong Hu, Ya-Ping Fang
In the light of the definitions above, Lipschitz continuity implies absolute continuity, which gives rise to uniform continuity. An absolutely continuous function is differentiable almost everywhere.If is absolutely continuous and is Lipschitz continuous with constant L, then their composition function is absolutely continuous. Moreover, F is almost everywhere differentiable and the inequality holds almost everywhere.
Tseng's extragradient algorithm for pseudomonotone variational inequalities on Hadamard manifolds
Published in Applicable Analysis, 2022
Jingjing Fan, Xiaolong Qin, Bing Tan
Motivated by the results described above, the aim of this paper is to present an extragradient algorithm for variational inequalities associated with pseudomonotone vector fields in Hadamard manifolds and to study the convergence properties of the extragradient algorithm. We first incorporate the Tseng's extragradient method with a suitable linesearch to remove the dependence on the Lipschitz continuity modulus of A when choosing stepsize λ. In particular, we weaken the Lipschitz continuity of A to the uniform continuity, which is crucial when the operator is not Lipschitz continuous or the Lipschitz modulus is difficult to estimate in advance. To the best of our knowledge, this result has not been studied in Hadamard manifolds before. It is worth mentioning that our results can be seen as a generalization of the corresponding results presented by Thong and Vuong [20] in real Hilbert spaces.
Modified Tseng's extragradient methods for solving pseudo-monotone variational inequalities
Published in Optimization, 2019
Duong Viet Thong, Phan Tu Vuong
The aim of this paper has two folds. We first incorporate the Tseng's extragradient method studied in [22,23] with a suitable linesearch to remove the dependence on the Lipschitz continuity modulus of A when choosing stepsize λ. We also weaken the Lipschitz continuity of A to the uniform continuity. This is crucial when the operator is not Lipschitz continuous and/or the Lipschitz modulus is difficult to estimate in advance. Doing so, we obtain the weak convergence of the iterative sequence. As we are working in infinite dimensional Hilbert spaces, the strong convergence is essential. Therefore, in the second part of the paper, we combine the line-search method with a Mann-type iteration step to obtain the strong convergence of the iterative sequence.