Spaces of functions on manifolds or spaces of maps of manifolds to, say, Rⁿ look rather similar to each other. They are infinite dimensional linear spaces and all their points are somewhat equivalent. Probably, I.Gelfand was the first who made the remark that, in order to preserve the information about the topology of the manifold, one should consider this space with the discriminant in it: the subset of singular objects. The topology of the discriminant (including the local one) is connected with invariants of maps. It led to the theory of Vassiliev invariants of knots (invariants of first order). We shall discuss how the topology of the discriminant permits to define and to construct, so-called, invariants of first order (the general setting belonging to V.Arnold and V.Vassiliev).