The following is taken from 4.1 in Liu's book.
Definition: A scheme $X$ is normal at $x \in X$ if $O_{X,x}$ is normal. $X$ is normal if it is irreducible and normal at every point.
Proposition: Let $X$ be irreducible. Then $X$ is normal if and only if for every $U \subseteq X$ open, $O_X (U)$ is a normal integral domain.
I have a question on the phrasing of this statement. Reviewing the proof, he seems to assume that $O_X (U)$ is already an integral domain. Of course, we wouldn't be talking about a ring being normal if it weren't an integral domain. However, the phrasing of the theorem seems to assert that $O_X (U)$ is an integral domain for every open subset $U$ as opposed to asserting that for the open subsets $U$ where $O_X (U)$ is an integral domain. Which is the correct way to interpret this Proposition?