In this chapter we return to unfoldings of functions and give a proof of the main theorem, 6.6p, for analytic functions and families. The work involved in this is quite substantial, and a good deal more complicated than anything else in the book. There is, however, a relatively short initial section of the proof which at least makes the result quite plausible and is relatively easy to follow. Before plunging into the proof we give some explanation of why we shall only deal with the analytic case.

As we have mentioned before, the complete proof of the main theorem 6.6p for smooth functions and families unfortunately requires a rather formidable technical result called the Malgrange preparation theorem. This result, together with Sard's theorem (4.18), forms the cornerstone of the theory of smooth maps, but a proof would be out of place in a book of this sort. (A complete proof, together with a proof of the main theorem on unfoldings, appears in Bröcker and Lander (1975).) What we do here is assume that all our functions are analytic, that is given by convergent power series, and produce analytic families of functions which induce any unfolding of tk+1 from the unfolding G of 6.6p. This involves two steps: producing the power series and proving that they are convergent.