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Docta Complutense author Beer, Gerald author Garrido Carballo, María Isabel 2023-06-18T05:40:36Z 2023-06-18T05:40:36Z 2015-08-04 Beer, Gerald, y M. Isabel Garrido. «Locally Lipschitz Functions, Cofinal Completeness, and UC Spaces». Journal of Mathematical Analysis and Applications, vol. 428, n.o 2, agosto de 2015, pp. 804-16. DOI.org (Crossref), https://doi.org/10.1016/j.jmaa.2015.02.085. 10.1016/j.jmaa.2015.02.085 https://hdl.handle.net/20.500.14352/22987 https://doi.org/10.1016/j.jmaa.2015.02.085 http://www.sciencedirect.comhttp://www.sciencedirect.com/science/article/pii/S0022247X15002139# 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]. eng Locally Lipschitz functions, cofinal completeness, and UC spaces journal article URL https://docta.ucm.es/bitstreams/fc7ffb65-f244-4eaf-ab6b-a4799a89a685/download File MD5 b06f8ba8330ded4266f10df73fad3e58 314971 application/pdf Garrido16.pdf URL https://docta.ucm.es/bitstreams/16616f3b-0c83-4122-a58c-8e9758b0255c/download File MD5 36d255f3d7d1eb31639dd7c8154852a7 410283 application/pdf Garrido17.pdf