Abstract: Processes for the preparation of a functional polymer of low molar mass, which includes initially charging a first quantity M1 of a mixture including at least one component selected from olefinically unsaturated monomers B and compounds A which are not themselves copolymerizable but which are able to react with at least one of the monomers B to form a copolymerizable addition compound, metering in prior to or during polymerization a second quantity M2 of a mixture whose composition can be identical to or different from that of the first mixture M1 and which includes at least one component selected from olefinically unsaturated monomers B and compounds A, initiating polymerization by use of a polymerization initiator and optionally a chain transfer agent, conducting polymerization in the presence of an oligomer or polymer C, wherein the oligomer or polymer C is either formed in situ from one or more of A and B or is dissolved in at least one of the quantities M1 and M2, wherein at least one of M1 and M2 includ