Let \(n\geq 1\), let \(f\colon S^n\to S^n\) be a continuous map, and let \(y\in S^n\) be a point whose preimage \(\inverseof{f}(y) = \ts{x_1, \ldots, x_m}\) is finite. Homology is taken with coefficients in \(\ZZ\). Choose an open neighborhood \(V\) of \(y\) and disjoint open neighborhoods \(U_i\) of \(x_i\) with \(f(U_i)\subseteq V\), and fix a generator of \(H_n(S^n)\). Then \(f\) restricts to maps of pairs \(f_i\colon (U_i, U_i\sm\ts{x_i})\to (V, V\sm\ts{y})\), and under the isomorphisms \[ H_n(U_i, U_i\sm\ts{x_i})\cong H_n(S^n, S^n\sm\ts{x_i})\cong H_n(S^n)\cong\ZZ, \qquad H_n(V, V\sm\ts{y})\cong H_n(S^n, S^n\sm\ts{y})\cong H_n(S^n)\cong\ZZ \] given by excision and the long exact sequence of the pair, the induced homomorphism \(f_{i*}\) is multiplication by an integer. This integer is the local degree \(\deg f\vert_{x_i}\) of \(f\) at \(x_i\).
Let \(n\geq 1\), let \(f\colon S^n\to S^n\) be a continuous map, and let \(y\in S^n\) be a point with finite preimage \(\inverseof{f}(y) = \ts{x_1, \ldots, x_m}\). Then \(\deg f = \sum_{i=1}^m \deg f\vert_{x_i}\), where \(\deg f\) is the degree of \(f\).
[@Hat02].