Let $f: X \to Y$ be a morphism between two quasiprojective, irreducible varieties over the complex numbers, such that the image of $f$ is Zariski dense in $Y$ and there is a Zariski dense subset $U$ (not necessarily open) of $X$ such that we have $\{x\}=f^{-1}(f(x))=\{y \in X: f(y)=f(x)\}$ for all $x \in U$.

Is $f$ then always a birational morphism? If not, is there a simple counterexample?

(Sorry for the first version of the question)