A class of functions called weakly convex and weakly quasiconvex functions is introduced by relaxing the definitions of convex and quasiconvex functions. Necessary and sufficient conditions, under which a lower semi-continuous function on a nonempty closed convex subsets of the n-dimensional Euclidean space is convex or quasiconvex are established. Another class of functions, called $-convex and $-B-vex, is defined. Some properties for these functions are presented.