It is known that the Brouwer degree for each continuous function \(f: \overline{\Omega} \to \mathbb R^N\) where \(\Omega\) is a bounded open set in \(\mathbb R^N\) can be extended to the case when \(\Omega\) is unbounded provided that \(f\) is proper. However, difficulties arise when \(\Omega\) is unbounded but \(f\) is not proper. In the present paper, a degree for every continuous function \(f: \mathbb R^N \to \mathbb R^N\) is defined
\[ \deg (f,y):= \deg(f,B_r,y) \quad \text{for \(r\) large enough} \]
where the degree in the right-hand side is Brouwer’s degree and \(B_r\) denotes the open ball in \(\mathbb R^N\) with center \(0\) and radius \(r\), provided that \(y\) is restricted to the complement of some closed subset \(A(f)\) of “asymptotic” values
\[ A(f):=\{y\in \mathbb R^N : y=\lim f(x_n) \text{ with }(x_n)\subset \mathbb R^N, \;\lim| x_n|=\infty\}. \]
This degree possesses some basic properties of the Brouwer degree. Moreover, sufficient conditions are given for \(A(f)\) to be nowhere dense, so that \(\deg(f,y)\) is defined for \(y\) in an open dense subset of \(\mathbb R^N\). It is also shown that, in the opposite direction, \(A(f)\) having nonempty interior has a direct impact on the solvability of the equation \(f(x)=y\), which cannot be discovered by degree arguments.