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Locally Lipschitz functions, cofinal completeness, and UC spaces

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Beer, Gerald and Garrido, M. Isabel (2015) Locally Lipschitz functions, cofinal completeness, and UC spaces. Journal of Mathematical Analysis and Applications, 428 (2). pp. 804-816.

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Official URL: http://www.sciencedirect.com/science/article/pii/S0022247X15002139#


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Abstract

Let X, d be a metric space. We find necessary and sufficient conditions on the space for the locally Lipschitz functions to coincide with each of two more restrictive classes of locally Lipschitz functions studied by several authors: the uniformly locally Lipschitz functions and the Lipschitz in the small functions. In the first case, we get the cofinally complete spaces and in the second, the UC spaces. We address this question: to which family of subsets of X, d is the restriction of each function in each class actually Lipschitz? Finally, we determine exactly when the class of uniformly locally Lipschitz functions is uniformly dense in the Cauchy continuous real-valued functions, a class that naturally contains them. In fact, our theorem is valid when the target space is any Banach space. Our density theorem complements the uniform approximation results of Garrido and Jaramillo [12, 13].


Item Type:Article
Uncontrolled Keywords:Lipschitz function, locally Lipschitz function, uniformly locally Lipschitz function, Lipschitz in the small function, Cauchy continuous function, uniform approximation, UC-space, cofinally complete space, uniformly paracompact space
Subjects:Sciences > Mathematics > Functional analysis and Operator theory
Sciences > Mathematics > Topology
ID Code:29162
Deposited On:11 Mar 2015 10:03
Last Modified:12 Dec 2018 15:12

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