# Semi-continuity in quasi-finite morphisms without properness

Let $$f:U \to V$$ be a flat, quasi-finite, surjective morphism between two affine varieties defined over $$\mathbb{C}$$. Assume that every closed fiber is reduced. Consider the function $$\eta$$ that sends a closed point $$v \in V$$ to the cardinality of the fiber $$f^{-1}(v)$$ over $$v$$. Is this function semi-continuous? If so, is it upper or lower semi-continuous? Recall, if we add the assumption that $$f$$ is proper, then it is well-known that $$\eta$$ is locally-constant.

Note that since you assume the fibres reduced hence (because the characteristic is 0) geometrically reduced, the map $$f$$ is in fact étale; and closed points have residue field $$\mathbb{C}$$ which is algebraically closed; hence your $$\eta(v)$$ is indeed the function $$n(v)$$ from EGA4.