  In this note we investigate two kinds of applications of the
Painlev\'e-Kuratowski convergence of closed sets in analysis that are motivated
also by questions from singularity theory. First, we generalise to Lipschitz
functions the classical theorem stating that given a sequence of smooth
functions with locally uniformly convergent derivatives, we obtain the local
uniform convergence of the functions themselves (provided they were convergent
at one point). Next we turn to the study of the behaviour of the fibres of a
given function. We prove some general real counterparts of the Hurwitz theorem
from complex analysis stating that the local uniform convergence of holomorphic
functions implies the convergence of their sets of zeros. From the point of
view of singularity theory our two theorems concern the convergence of the sets
when their descriptions are convergent. They are also of interest in
approximation theory.
