Pole divisor avoiding a fiber

Suppose a nonconstant morphism of varieties $f:X\to Z$ has fiber $Y$ over $z_0$, where $X$ is normal, connected and projective. Choose an affine neighborhood $U$ of $z_0$ and a regular function on $U$ nonconstant on $f(X)\cap U$. Its pullback is a nonconstant rational function on $X$, so its pole Weil divisor is nonzero. A pole-free rational function on a normal variety is regular, and global regular functions on a connected projective variety are constant. The pole divisor avoids $f^{-1}(U)$, hence avoids $Y$. No projectivity assumption on $Z$ is needed.