On the Multiplicity of a C^∞-differentiable Function-germ

Risler ＆ Trotman in 1997 proved that the multiplicity of an analytic function germ is left-right lipschitz invariant, which provided a partial answer to Zariski conjecture. In this note, based on the recent work of Comte, Milman ＆ Trotman, we generalize the work of them to prove that the multiplicity of a C^∞ differentiable function germ is also left-right lipschitz invariant.