= Solution
For closed connected oriented $d$-manifolds, the <degree of a map between oriented manifolds> is the integer defined by $f_*[M]=(\deg f)[N]$ in $H_d(N;\mathbb Z)$. If $y\in N$ has finite fiber $\{x_1,\ldots,x_r\}$, choose disjoint small neighbourhoods $U_i$ containing no other points of that fiber. The map of local <relative homology> groups
$$
H_d(U_i,U_i\setminus\{x_i\};\mathbb Z)\longrightarrow H_d(N,N\setminus\{y\};\mathbb Z)
$$
sends the local orientation generator at $x_i$ to $d_i$ times the one at $y$. This integer is its <local degree of a continuous map>, independent of the small neighbourhood by <excision>. Excision also identifies $H_d(M,M\setminus f^{-1}(y))$ with the direct sum of the local groups, and the global fundamental class restricts to their orientation generators. Naturality of the maps to relative homology therefore proves the <degree as a sum of local degrees>:
$$
\boxed{\deg f=\sum_{i=1}^r\deg_{x_i}f.}
$$
The empty fiber gives degree zero. For smooth maps at a regular value, a local degree is the sign of the derivative determinant, but general continuous maps can have larger local multiplicities.
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