  In this paper, by using functor rings and functor categories, we study
finiteness and purity of subcategories of the module categories. We give a
characterisation of contravariantly finite resolving subcategories of the
module category of finite representation type in terms of their functor rings.
We also characterize contravariantly finite resolving subcategories of the
module category $\Lambda${\rm -mod} that contain the Jacobson radical of
$\Lambda$ of finite type, by their functor categories. We study the pure
semisimplicity conjecture for a locally finitely presented category
${\underrightarrow{\lim}}\mathscr{X}$ when $\mathscr{X}$ is a covariantly
finite subcategory of $\Lambda$-mod and every simple object in
Mod$(\mathscr{X}^{\rm op})$ is finitely presented and give a characterization
of covariantly finite subcategories of finite representation type in terms of
decomposition properties of their closure under filtered colimits. As a
consequence we study finiteness and purity of $n$-cluster tilting subcategories
and the subcategory of the Gorenstein projective modules of the module
categories. These results extend and unify some known results.
