Global inverse function theorems

It is often of interest to determine whether it is possible to invert a given mapping
f
of one subset of
R^n
into another. This paper reports on a collection of pertinent results and related material. We first prove a theorem which is used in latter sections of the paper. It provides several characterizations of a homeomorphism between open subsets of
R^n
. Corresponding results are then proved for
C^k
-diffeomorphisms Another theorem, one of our main results, shows that a
C^k
map
f
between two spaces of a certain general type is a
C^k
-diffeomorphism if and only if a certain steepest descent process can always be carried out, and for each
y
converges globally to a unique solution
x
of
f(x)=y
. A final section is concerned with specific results for mappings that take one open rectangular region in
R^n
into another. These results are of direct interest, for example, in the area of economics, where they provide definitive invertibility conditions for a large class of price-demand relations.