This question was asked in my complex analysis quiz and I was unable to solve it . So I am asking for help here.
Question: Give an example of a bounded holomorphic function f on $\mathbb{C}$/$\mathbb{R}$ which cannot be extended to a larger set.
I am sorry but I can't provide any meaningful and an attempt in the direction of this question as I am really confused by this peciluar question.
It is my humble request to help me with this question.
Take $f = 1$ in the upper half plane and $f = -1$ in the lower half plane.