  We deal with non-smooth differential systems $\dot{z}=X(z), z\in R^{n},$ with
discontinuity occurring in a codimension one smooth surface $\Sigma$. A
regularization of $X$ is a 1-parameter family of smooth vector fields
$X^{\delta},\delta>0$, satisfying that $X^{\delta}$ converges pointwise to $X$
in $R^{n}\setminus\Sigma$, when $\delta\rightarrow 0$. We work with two known
regularizations: the classical one proposed by Sotomayor and Teixeira and its
generalization, using non-monotonic transition functions. Using the techniques
of geometric singular perturbation theory we study minimal sets of regularized
systems. Moreover, non-smooth slow-fast systems are studied and the persistence
of the sliding region by singular perturbations is analyzed.
