Question

We just consider puntured unit disk $\triangle^{*}$ in $\mathbb{C}$. $f$ is a bounded quasiconformal map on $\triangle^{*}$. Why $f$ can extend to the origin,becoming quasiconformal map on the whole disk?

rk:It is easy to see we only have to show that we can define $f(0)$ such that $f$ is continous.Then quasiconformality of new $f$ follows easily!

Answer 1

A simple way is the use the modulus of curve family argument, see Väisälä's book lecture notes on n-dimensional quasiconformal mappings.