In this paper we generalize the convex functions, defining the concept of
preconvex function and we study some characterizations by intervals, some characterizations by polytopes, some characterizations by level sets, some properties of the extreme points and some relations whith the convex functions. Also, we define the R-quasiconvex functions as a generalization of the quasiconvex functions, and we study some characterizations by level sets and by separation sets, and some relations with the quasiconvex functions
