I'd like to see a complete proof of the simplest version of the following rough statement: "If $f/g$ is a rational function on a reduced scheme ($g$ not a zero divisor), and $f/g$ doesn't have poles in codimension $1$, then $f/g$ is a well-defined function on the normalization."

I figure this should be called the valuative criterion of integrality (and if not, then maybe that's my problem). I can't find it in [Eisenbud], [Vakil], [Hartshorne], or [Vasconcelos]. There is a version of it in [Huneke-Swanson], but they're treating the more general case of integral closure of ideals than merely of rings, which obviously I should be able to downgrade from but would rather not if I don't have to. Where should I read about it?