Propylene-ethylene copolymer (PER), atactic polypropylene (PP) and atactic poly-4-methyl-1-pentene (P4MP1) were prepared with a Cp2ZrCl2-methylaluminoxane catalyst system. These polymers containing terminal vinylidene groups were made to react with halogen, sulfuric acid, 9-borabicyclo [3,3,1] nonane (9-BBN) or percarboxylic acid, leading to terminally halogenated, hydrogen-sulfated, hydroxy-(OH)- or epoxy- functionalized polyolefines.