Is the restriction of a finite map of affine varieties also finite?

Regular map $f:X\to Y$ of affine varieties correspond bijectively to morphisms of $k$-algebras $\phi: k[Y]\to k[X]$.
The map $f$ is dominant iff $\phi$ is injective, and is (by definition) finite iff $\phi$ makes $k[X]$ a finite $k[Y]$- module i.e. if $k[X]$ is a module of finite type over $\phi(k[Y])$ .
[The definition in Shafarevich is equivalent but confuses the issue with irrelevant integrality conditions.]
For any subset $Z\subset X$ the induced morphism $\bar \phi:k[Y]/\phi ^{-1}(I(Z))\to k[X]/I(Z) $ is also injective and module-finite so that the corresponding geometric restriction map map $\operatorname {res}(f): Z\to \overline {f(Z)}$ is dominant and finite, just as $f$ was.

NB
Notice that, in conformity with your request, I have never mentioned closedness of any map.