Finite $C^0$-determinacy of real analytic map germs with isolated instability

Abstract

Let $f\colon(\mathbb R^n,0)\to(\mathbb R^p,0)$ be a real analytic map germ with isolated instability. We prove that if $n=2$ and $p=2,3$, then $f$ is finitely $C^0$-determined. This result can be seen as a weaker real counterpart of Mather-Gaffney finite determinacy criterion.