To easily obtain functional polymers containing a low boiling point monomer at a high ratio by copolymerizing an olefinically unsaturated monomer with a component capable of reacting with this monomer to form a polymerizable addition compound in a specific manner.

A first quantity M1 of a mixture containing at least one component selected from olefinically unsaturated monomers B and a component A which is not itself copolymerizable but which is able to react with at least one of monomers (B) to form a copolymerizable addition compound is initially charged prior to the initiation of polymerization. Then, polymerization is initiated by the addition of a polymerization initiator and, optionally, a molecular weight modifier. During polymerization, another quantity M2 of a mixture whose composition may be identical to or different from that of the first mixture M1 and which contains at least one component selected from olefinically unsaturated monomers B and compounds A is metered and supplied. In this instance, an oligomer or polymer C is dissolved in at least one of the quantities M1 and M2.