Every bounded holomorphic function on $\mathbb{C}\backslash \{0\}$ must be constant

Question

I'm trying to prove this via Liouvilles theorem but not sure how the exclusion of this point affects the conclusion of the theorem. We know nothing about the behaviour at 0 so extending it to complex plane may not necessarily work as it may not be bounded or holomorphic on at the point 0. Can anyone shed some light on this? maybe it's not even true?