The classical implicit function theorem states that if X, Y , and Z are Banach spaces and F is a continuously differentiable mapping from a neighborhood of a point (x, y) ∈ X × Y into Z such that F(x, y)=0 and Fy(x, y) is an invertible operator, then there exists a continuously differentiable mapping (implicit function) ϕ from a neighborhood of V of the point x into Y for which the equality F(x, ϕ(x)) = 0 holds for all x ∈ V . It turns out that, under these assumptions, more is true: there exists an implicit function not only for the mapping F, but also for all mappings that are close enough (in a certain sense) to F. Moreover, if a mapping close to F is continuously differentiable, then the corresponding implicit function is also continuously differentiable. Such properties are important for applications in situations where the source data is not precise or where an “intricate” mapping F is approximated by a simpler one.