 Resolvent propagation across a complementary component

The polynomially approximable resolvent points form a subset that is both open and closed relative to $\mathbb{C}\setminus K$. Openness follows from the uniformly convergent expansionwhen $|\mu -\lambda |<\mathrm{dist}(\lambda ,K)$; relative closedness follows from the uniform continuity of the resolvent as its pole varies away from $K$. Hence approximability at one point propagates throughout its connected component of $\mathbb{C}\setminus K$.