Let $f:Y\to X$ be a surjective finite morphism between Noetherian schemes. The definition of a finite morphisms requires $f^\#_U:\mathcal{O}_X(U)\to \mathcal{O}_Y(f^{-1}(U))$ to be a finite ring homomorphism, for every affine open $U\subset X$.
Does that imply $f^\#_U:\mathcal{O}_X(U)\to \mathcal{O}_Y(f^{-1}(U))$ is a finite ring homomorphism for an arbitrary open $U\subset X$, and in particular for $U=X$? If not, is there some mild conditions under which these homomorphism become finite?
References or counterexamples would be appreciated.