Let $r: X\to Y$ be a morphism of algebraic sets (over $\mathbb C$) and let $X'$ be an algebraic subset of $X$ containing $x_0.$ It is easy to see that if $r_*: T_{x_0} X\to T_{r(x_0)} Y$ is a monomorphism then its restriction to $T_{x_0} X'\to T_{r(x_0)}\, r(X')$ is a monomorphism as well. But does $r_*$ being an epimorphism imply that its restriction to $T_{x_0} X'\to T_{r(x_0)}\, r(X')$ is an epimorphism as well?
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