Magnification Spaces: A nonstandard approach to inverse mapping theorems

Abstract

This paper develops an infinitesimal order of magnitude coupled with overflow technique that allows nonnumerical proofs of nondegenerate and degenerate inverse mapping theorems for mappings minimally regular at a point. This approach is used first to give a transparent proof of the inverse mapping theorem of Behrens and Nijenhuis and then is deployed to prove an inverse mapping result for mappings whose linear part vanishes at the differentiable point. We finish by indicating further possible capacities of this approach.