AbstractLetGbe a locally compact commutative group and letgandhbe positive definite functions onG, which are not identically zero. We show that continuity ofgh̄implies the existence of a character γ ofGd(the discrete version ofG) such that γgand γhare continuous. As corollary we get a special case of a result of K. de Leeuw and I. Glicksberg concerning almost continuous group representations. In the second part of the paper we prove decomposition theorems for positive definite functions defined on a neighbourhood of the z