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∈ℂn 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 μ*-sequence). We show that if f and f′ can be connected by a continuous path in the μ-constant (respectively, μ*-constant) stratum, then they can also be connected by a piecewise complex-analytic path (defined in the relevant natural way) in this stratum.