TITLE: Is the restriction of a finite map of affine varieties also finite?
QUESTION [7 upvotes]: If $f:X\to Y$ is a dominant(i.e.$f(X)$ is dense in $Y$) regular map of affine varieties, then $f$ is called a finite map if $k[X]$ is integral over $f^*k[Y]$.

My question is: if $Z\subset X$ is a closed subset of $X$, then how to
  show the restiction $f|_Z: Z\to \overline{f(Z)}$ is still a finite
  map?

(Note: This result is supposed to prove the fact that "a finite map takes closed sets to closed sets"(cf.Page60, Basic Algebraic Geometry 1, by Shafarevich), so please do not use the latter fact in the answer.)

REPLY [9 votes]: 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.