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On the uniform approximation of Cauchy continuous functions Beer, Gerald Garrido Carballo, María Isabel 517.53 Cauchy continuous function Cauchy-Lipschitz function Lipschitz in the small function Locally Lipschitz function Primary Secondary Uniform approximation Funciones (Matemáticas) 1202 Análisis y Análisis Funcional 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. Ministerio de Economía y Competitividad (España) Depto. de Álgebra, Geometría y Topología Fac. de Ciencias Matemáticas TRUE pub 2023-06-18T06:54:06Z 2023-06-18T06:54:06Z 2016 journal article https://hdl.handle.net/20.500.14352/24538 0166-8641 10.1016/j.topol.2016.04.017 eng info:eu-repo/grantAgreement/MINECO//MTM2012-34341/ES/ Beer, Gerald, y M. Isabel Garrido. «On the Uniform Approximation of Cauchy Continuous Functions». Topology and Its Applications, vol. 208, agosto de 2016, pp. 1-9. DOI.org (Crossref), https://doi.org/10.1016/j.topol.2016.04.017 restricted access application/pdf Elsevier Science