If $f:\mathbb{C}\backslash S^1\rightarrow \mathbb{C}$ is holomorphic where $S^1=\{z\in \mathbb{C}:|z|=1\}$ and $f$ continuous in all $\mathbb{C}$ then $f$ is holomorphic in all $\mathbb{C}$.
All the results I know to prove that is holomorphic considers isolated singularities. I think that writing $$f(z)=\sum_{n=0}^\infty a_n z^n,|z|<1$$ might help. Because if I can prove that $$f(z_0)=\sum_{n=0}^\infty a_n z_0^n,|z_0|=1$$ then I finish. Right know the best I know is $f(z_0)=\lim_{z\rightarrow z_0}\sum_{n=0}^\infty a_n z^n$ taking the limit with $|z|<1$.