All the continuation theorems presented so far were global, even if, in some cases, some local arguments have been used in their proofs.

For a given map : V C & X [ O , 1 ] -+ O, a subset B C O and an element (xo, Q) E V with (xo, 0) E B, we have tried to prove the existence of a map

and the corresponding existence result is said to be a (global) continuation or (global) implicit function theorem. In many cases it is important to know if the implicit function is unique and has some additional properties. For example, when E is a set in a normed space, we can ask about the existence of a continuous or differentiable implicit function. Such properties of the implicit function are reflections of the corresponding properties of 7.