understanding the definition of domains of holomorphy

Question

can anyone give me an example and explain why any open set in $\mathbb{C}$ is a domain of holomorphy?

I have understood the fact from here but not able to understand their explanation for $n=1$

Answer 1

It's clear that $\mathbb{C}$ itself is a domain of holomorphy.

For other domains, let $p$ be a boundary point of $\Omega$ and put $$f(z) = \frac{1}{z-p}.$$ Then $f$ is holomorphic on $\Omega$ (indeed on $\mathbb{C} \setminus \{ p \}$) and can't be extended across $p$.