Let $f:X \rightarrow Y$ be a morphism of algebraic varieties over an algebraically closed field. If all fibers $f^{-1}(y)$ with $y$ closed point in $Y$ are finite, can one conclude that an arbitrary fiber (i.e. with $y$ not necessarily closed point) is finite?
Edit: By a fiber being finite I just mean it to consist of a finite number of points.