We show that every μ-constant family of isolated hypersurface singularities of type F(x, t) =f(x)+tg(x), where t is a parameter, is topologically trivial. The proof uses only the curve selection lemma, and hence, for an appropriately translated statement, also works over the reals and for some families of non-isolated singularities. Some applications to study the singularities at infinity of complex polynomials are given.