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On the uniform approximation of Cauchy continuous functions

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Beer, Gerald and Garrido, M. Isabel (2016) On the uniform approximation of Cauchy continuous functions. Topology and its Applications, 208 (1). pp. 1-9. ISSN 0166-8641

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Official URL: http://www.sciencedirect.com/science/article/pii/S0166864116300578


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Abstract

In the context of real-valued functions defined on metric spaces, it is known that the locally Lipschitz functions are uniformly dense in the continuous functions and that the Lipschitz in the small functions - the locally Lipschitz functions where both the local Lipschitz constant and the size of the neighborhood can be chosen independent of the point - are uniformly dense in the uniformly continuous functions. Between these two basic classes of continuous functions lies the class of Cauchy continuous functions, i.e., the functions that map Cauchy sequences in the domain to Cauchy sequences in the target space. Here, we exhibit an intermediate class of Cauchy continuous locally Lipschitz functions that is uniformly dense in the real-valued Cauchy continuous functions. In fact, our result is valid when our target space is an arbitrary Banach space.


Item Type:Article
Uncontrolled Keywords:Cauchy continuous function; Cauchy-Lipschitz function; Lipschitz in the small function; Locally Lipschitz function; Primary; Secondary; Uniform approximation
Subjects:Sciences > Mathematics > Functions
ID Code:38178
Deposited On:23 Jun 2016 08:23
Last Modified:12 Dec 2018 15:12

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