I'm reading Milnor's book "Topology from the differential viewpoint" and I'm stuck at this point:
Let $M$ and $N$ be oriented n-dimensional manifolds without boundary and let $$f: M \longrightarrow N$$ be a smooth map. If $M$ is compact and $N$ is connected it is asserted that the degree of $f$, $\deg(f,y)= \sum_{x \in f^{-1}(y)}\mathrm{sign} \ \mathrm{d}f_x$, is a locally constant function of $y$, as $y$ ranges through regular values. I'm not sure how to prove it but I have thought this:
We know that $\# f^{-1}(y)$ is locally constant and that the sign of $\det(\mathrm{d}f)$ (i.e $\mathrm{sign} \ \mathrm{d}f$) is locally constant too. So I guess that we can conlude the desired result from this. But I'm a little bit confused because I didn't use the fact that $N$ is connected... Any ideas? Thanks a lot.