Local continuous extension of proper holomorphic maps: low-regularity and infinite-type boundaries

We prove a couple of results on local continuous extension of proper holomorphic maps $F:D\to \mathrm{\Omega }$, $D,\mathrm{\Omega }⊊{\mathbb{C}}^n$, making local assumptions on $\mathrm{\partial }D$ and $\mathrm{\partial }\mathrm{\Omega }$. The first result allows us to have much lower regularity, for the patches of $\mathrm{\partial }D,\mathrm{\partial }\mathrm{\Omega }$ that are relevant, than in earlier results. The second result (and a result closely related to it) is in the spirit of a result by Forstneric–Rosay. However, our assumptions allow $\mathrm{\partial }\mathrm{\Omega }$ to contain boundary points of infinite type.