Extending homeomorphism of unit circle to unit disk

Question

What is the best best way to prove that any homeomorphism of the unit circle onto itself can be extended to a homeomophism of the closure of the unit disk onto itself?

Answer 1

Let $f:\left\{ z\in\mathbb{C}:\left|z\right|=1\right\} \rightarrow\left\{ z\in\mathbb{C}:\left|z\right|=1\right\} $ be a homeomorphism.

Then $g:\left\{ z\in\mathbb{C}:\left|z\right|\leq1\right\} \rightarrow\left\{ z\in\mathbb{C}:\left|z\right|\leq1\right\} $ defined by $0\mapsto0$ and $z\mapsto\left|z\right|f\left(\frac{z}{\left|z\right|}\right)$ if $z\neq0$ is a homeomorphism of the closure of the unit disk unto itself, and extends $f$. I don't know whether it deserves the qualification 'the best'. Maybe more something as 'most natural'.