This is inspired by https://math.stackexchange.com/questions/705979/purely-inseparable-morphisms-and-factorizations-of-a-morphism-of-finite-type that I asked previously .
Let $k$ be an algebraically closed field. Let $f: X \rightarrow Y$ be a finite surjective morphism of integral algebraic varieties over k. Suppose we know the following:
If $K(X)/K(Y)$ is separable of degree n, then there exists a non-empty open subset V of Y such that for any $y \in V(k)$, $f^{-1}(y)$ consists of at least n points.
Then, with this I want to show that if f induces an injective map $X(k) \rightarrow Y(k)$ then it is purely inseparable. I see that one should try to apply the above, but I can't quite make it work - what if the extension $K(X)/K(Y)$ is a mix between being separable and inseparable for example?