The study of bifurcations in a piecewise-smooth system is quite different from those occurring in smooth systems. In piecewise-linear systems, which we are considering in this paper, mainly border collision bifurcations (BCBs) and contact bifurcations occur. It is classified as border-collision any contact between an invariant set of a map with the border of its region of definition, and this may, or may not, produce a bifurcation. The term border-collision bifurcation was introduced for the first time by Nusse and Yorke in 1992 and is now widely used in this context (i.e. for piecewise smooth maps). Recently, these bifurcations have been studied mainly because of their relevant applications in engineering (electrical and mechanical), and also in economics and social sciences. In fact, several publications were motivated by models describing particular circuits or models for the transmission of signals. Remarkably also before the work by Nusse and Yorke the bifurcations associated with piecewise smooth maps were studied in some publications (although the bifurcations were not called border-collision bifurcations). We recall, for example, the works by Mira (1978) and others. We may also go further back, citing the work by Leonov in the end of 50th. In his work, Leonov described several bifurcations, giving a recurrence relation to find the analytic expressions of families of bifurcations occurring in a one-dimensional piecewise linear map with one point of discontinuity, which is still mainly unknown. The object of this work is to give a new interpretation and improvements of some of his results.