This paper concerns differential equation systems on R(n) with discontinuous right-hand sides. We deal with non-smooth vector fields in R(n) having a codimension-one submanifold M as its discontinuity set. After a regularization of a such system and a global blow-up we are able to bring out; some results that bridge the space between discontinuous systems and singularly perturbed smooth systems.Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES)Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq)Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP)