We suppose throughout that and are connected closed oriented manifolds of the same dimension . The degree of a continuous function is frequently computed according to the following considerations:
The space of continuous functions has a dense subspace consisting of smooth functions , and in particular every continuous function is homotopic to a smooth function . It therefore suffices to compute the degree of .
By Sard's theorem, the set of singular values? of a smooth function has measure zero (using for example the orientation on to define a volume form and hence a measure). Accordingly, we may choose a regular value? .
The inverse image is a compact -dimensional manifold, hence consists of finitely many (possibly zero) points . Since these are regular points, restricts to a diffeomorphism
where is a small neighborhood of and is a small neighborhood of . The diffeomorphism either preserves or reverses the orientation of , i.e., the sign of the determinant as a mapping between differential n-forms
is either or .
By a straightforward application of the excision axiom? in homology, it follows that the degree of is the sum of these signs:
and this quantity is independent of the choice of regular value .