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.