Brouwer’s degree without properness

A degree deg(f, y) is defined for every continuous function f :RN →RN, which possesses all the properties of Brouwer’s degree provided that y is restricted to the complement of some closed set A(f ) of “asymptotic” values. Sufficient conditions are given for A(f ) to be nowhere dense. It is also shown that, in the opposite direction, A(f ) having nonempty interior has a direct impact on the solutions of f (x) = y, which cannot be discovered by degree arguments. © 2007 Elsevier Inc. All rights reserved.