  Around 1970, Mather established a significant theory on the stability of
$C^\infty$ mappings and gave a characterization of the density of proper stable
mappings in the set of all proper mappings. The result yields a
characterization of the density of stable mappings in the set of all mappings
in the case where the source manifold is compact. The aim of this paper is to
complement Mather's result. Namely, we show that the set of stable mappings in
the set of all mappings is never dense if the source manifold is non-compact.
Moreover, as a corollary of Mather's result and the main theorem of this paper,
we give a characterization of the density of stable mappings in the set of all
mappings in the case where the source manifold is not necessarily compact.
