Lipschitz perturbations of differentiable implicit functions

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.