Want to extend a homeomorphism of the circle $S^1$ to the whole disk $D^2$ ?

Let $f\colon S^1\to S^1$ be a homeomorphism. Then the formula $$F(x)=||x||f(x/||x||)$$ allows you to define a map $F\colon D^2\to D^2$ which extends $f$ , for if $x\in S^1\subset D^2$ then $||x||=1$ and $F(x)=1\cdot f(x/1)=f(x)$ . Clearly this map is continuous, save (maybe) the origin, since this formula is undefined there. Nevertheless this is removable.

To check continuity at the origin use: ``A map $f$ is continuous at a point $p$ if and only if for each sequence $x_n\to p$ , $f(x_n)\to f(p)$ ''.

So take a sequence $u_n\in D^2$ such that $u_n\to 0$ (i.e. which tends to the origin). Then $F(u_n)=||u_n||f(u_n/||u_n||)$ and since $f(u_n/||u_n||)\neq 0$ , hence $||u_n||\to 0$ implies $F(u_n)\to 0$ , that is $F$ is also continuous at the origin.

The same method works for $f^{-1}$ .

In the same vein one can extend homeomorphisms $S^n\to S^n$ to $D^{n+1}\to D^{n+1}$ .