On the Domain of the Implicit Function and Applications

The Implicit Function Theorem asserts that there exists a ball of non zero radius within which one can express a certain subset of variables, in a system of equations, as functions of the remaining variables. We derive a lower bound for the radius of this ball in the case of Lipschitz maps. Under a sign preserving condition, we prove that an Implicit Function exists in the case of a set of inequalities. Also in this case, we state an estimate for the size of the domain. An application to the local Lipschitz behaviour of solution maps is discussed.