On paths in the μ-constant and μ*-constant strata

Let f and f' be non-constant, complex, polynomial function-germs at the origin 0 is an element of C-n such that f(0) = f '(0) = 0. We suppose that f and f' have an isolated singularity at 0 and the same Milnor number (respectively, the same Teissier ,mu*-sequence). We show that if f and f' can be connected by a continuous path in the ,mu-constant (respectively, mu*-constant) stratum, then they can also be connected by a piecewise complex-analytic path (defined in the relevant natural way) in this stratum.