The paper shows that Lipschitz and differentiability properties of a solution to a parameterized generalized equation 0∈f(x,y)+F(x), where f is a function and F is a set-valued map acting in Banach spaces, are determined by the corresponding Lipschitz and differentiability properties of a solution to z∈g(x)+F(x), where g strongly approximates f in the sense of Robinson. In particular, the inverse map (f+F)’-1 has a local selection which is Lipschitz continuous near x0 and Fréchet (Gateaux, Bouligand, directionally) differentiable at x0 if and only if the linearization inverse (f(x0)+∈f(x0)ë-x0)+F(ë))’-1 has the same properties. As an application, the paper studies directional differentiability of a solution to a variational inequality.
