In Kollár's article *The structure of algebraic threefolds: an introduction to Mori's program* he gives the following definition of a normal variety:

Definition 5.4.Let $V \subset \mathbb{C}^n$ be an algebraic variety and $v \in V$. We say $V$ isnormalat $v$ if every rational function bounded in some neighborhood of $v$ is regular at $v$.

This is in contrast to the definition of normal which I have seen everywhere else: the point $v \in V$ is normal if the local ring $\mathcal{O}_{V,v}$ is integrally closed. Of course the first question that came to mind is whether these two concepts are equivalent. I intuitively see that if for the rational function $g/h$ we have that $|(g/h)(w)| \to \infty$ as $w \to v$, then said function cannot satisfy a monic polynomial relation with regular coefficients (although I am not 100% sure how to prove this). However the converse seems very misterious to me. Is this the way to prove they are equivalent or am I following the wrong path? Are they equivalent at all?