  Let $y=f(x)$ be a continuously differentiable implicit function solving the
equation $F(x,y)=0$ with continuously differentiable $F.$ In this paper we show
that if $F_\eps$ is a Lipschitz function such that the Lipschitz constant of
$F_\eps-F$ goes to 0 as $\eps\to 0$ then the equation $F_\eps(x,y)=0$ has a
Lipschitz solution $y=f_\eps(x)$ such that the Lipschitz constant of $f_\eps-f$
goes to 0 as $\eps\to 0$ either. As an application we evaluate the length of
time intervals where the right hand parts of some nonautonomous discontinuous
systems of ODEs are continuously differentiable with respect to state
variables. The latter is done as a preparatory step toward generalizing the
second Bogolyubov's theorem for discontinuous systems.
