Beer, GeraldGarcía-Lirola, Luis C.Garrido, M. Isabel2023-06-172023-06-172020-04-251578-730310.1007/s13398-020-00847-xhttps://hdl.handle.net/20.500.14352/7724"This is a post-peer-review, pre-copyedit version of an article published in Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales. Serie A. Matemáticas. The final authenticated version is available online at: http://dx.doi.org/10.1007/s13398-020-00847-x”.This article provides necessary and sufficient conditions on the structure of a metric space such that for various vector lattices of real-valued Lipschitz-type functions defined on the metric space, the vector lattice is stable under pointwise product, and such that the reciprocal of each non-vanishing member of the vector lattice remains in the vector lattice. In each case the family of metric spaces for which the first property holds contains the family of metric spaces for which the second property holds. At the end we prove some extension theorems for classes of locally Lipschitz functions that complement known results for Cauchy continuous functions and for uniformly continuous functions.engjournal articlehttps://doi.org/10.1007/s13398-020-00847-xhttps://link.springer.com/article/10.1007/s13398-020-00847-xopen access515.122Lipschitz functionLocally Lipschitz functionLipschitz in the small functionCauchy-Lipschitz functionPointwise productReciprocationUC-spaceCofinal completenessModulus of continuityAnálisis funcional y teoría de operadoresFunciones (Matemáticas)Topología1202 Análisis y Análisis Funcional1210 Topología