Raw question:
Let $\Omega \subset \mathbb{C}$ be an open set with $z_{0} \in \partial \Omega$ and let $f$ be an analytic function on $\Omega$. How should one define, if it exists, the limit $\lim_{z\rightarrow z_{0}} f(z)$? I believe the answer is the usual definition with the understanding that $z \in \omega$, for all closed subsets $\omega \subset \Omega \cup \{z_{0}\}$
Additional remark:
I'm not convinced that the answer should be the usual definition with the understanding that $z$ belongs to a path $\gamma \subset \Omega$ which terminates at $z_{0}$ because $f(z)$ may have singularities on $\partial \Omega\backslash\{z_{0}\}$ and the path $\gamma$ may approach these singularities much faster than it approaches $z_{0}$.