A way to show that the Brouwer degree is constant on connected components

Question

Let $M, N$ be two oriented manifolds and let $A \subset M$ be an open subset s.t. $\overline{A}$ is compact. Let $\phi: \overline{A} \to N$ be continuous and smooth on $A$. For $y \in N \backslash \phi(\partial A)$, we have defined the degree of $\phi$ at each regular value $y$ as $$ d(f,y) = \sum_{x \in \phi^{-1}(y)} \text{sgn}(df_x). $$ How would one go ahead and prove that the degree is constant on connected components of $N \backslash \phi(\partial A)$.