Let $D$ be a bounded, finitely connected domain in $\mathbb{C}$ without isolated points in the boundary and let $f$ be a continuous function on $bD$. Let $\tilde{f}$ be a continuous extension of $f$ to $\bar{D}$. We prove that $f$ extends holomorphically through $D$ if and only if the degree of $\tilde{f}+h$ is non-negative for every holomorphic function $h$ on $D$ such that $\tilde{f}+h$ is bounded away from $0$ near $bD$.