I have to prove that every bounded function holomorphic on $\mathbb{C}^2 \setminus K$ is constant, where $K$ is
$(a)$ a ball
$(b)$ a complex line
$(c)$ an arbitrary analytic subset
Now, I think the idea here is to show that the holomorphic function can be extended to all of $\mathbb{C}^2$ and thereafter using Liouville's theorem to show the bounded function as constant, but I am kind of lost on how to proceed to do that. Can I get some help?
(i) If $K$ is a ball, we may use Hartogs' extension theorem.
(ii) Note a complex line is also an analytic subset. We may just consider the case $K$ is an analytic subset. But then $\mathbb{C}^2\setminus K$ is connected, and every locally bounded holomorphic functions on $\mathbb{C}^2\setminus K$ can be extended to $\mathbb{C}^2.$ See Demailly, complex analytic and differential geometry, P91 Remark 4.2.