Brouwer's degree without properness

A degree deg(f,y) is defined for every continuous function , 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.