Set-Valued Analysis by Covering

Here is a presentation of an answer to the problem, deeply studied in the 70ies and 80ies and to a large extent unsolved then, of differential and integral calculus transfer to the framework of set-valued analysis. This transfer is achieved through the identification of the set-valued maps with some families, the coverings, made up of some (class of) special functions, the representations. The process is related to the one of atlas local maps of differential geometry, see e.g. M. Demasure ["Catastrophes et Bifurcation", Ellipse 1989] and L. Schwartz ["Cours d'analyse", Ed. Hermann 1967]