In this MO question Terrence Tao writes the following about the inverse function theorem.
The Brouwer fixed point theorem gives local surjectivity, and degree theory gives local injectivity if $\det Df(x_0)$ never changes sign. (This gives another proof in the case when $f$ is continuously differentiable, since $\det Df$ is then continuous.)
I am not familiar with degree theory, so I cannot fill in the gaps here.
- How exactly does degree theory give locally injectivity if $\det Df(x_0)$ never changes sign?
- How exactly does the Brouwer fixed point theorem give local surjectivity?