The so-called Continuity Method is a simple yet powerful method to show that a given continuous injective map is surjective. Namely, suppose that $f:X \to Y$ is a map between two connected manifolds $X,Y$ of the same dimension. In order to show that $f(X)=Y$, it suffices to prove that
(i) $f:X \to Y$ is continuous and injective
and
(ii) $f(X)$ is closed.
This follows directly from Brouwer's Invariance of domain theorem.
This method was used by Koebe to prove that every multiply connected domain in the complex plane can be conformally mapped onto a circle domain. It seems to be sometimes referred to as the 'Koebe continuity method' or 'Brouwer-Koebe continuity method'.
My question is the following:
Question What are other examples of applications of the Continuity Method? Has it been applied in other areas than complex analysis?
Thank you,
Malik
