A detail in a proof that a scheme is integral if and only if it is reduced and irreducible

Question

I am confused about a small part of the following proof given in Hartshorne's Algebraic Geometry.

In particular, why do we have that the union of $Y$ and $Z$ is $U$?

Answer 1

Because $fg=0$, so in each fiber $O_x$, $x \in U$, $f_xg_x=0 \in m_x$. But $O_x$ is a local ring with $m_x$ its maximal ideal, so $f_x \in m_x$ or $g_x \in m_x$