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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!

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More homework than research, voting to close. – Igor Rivin Nov 19 '11 at 12:04

A simple way is the use the modulus of curve family argument, see Väisälä's book lecture notes on n-dimensional quasiconformal mappings. (Thm. 17.3)

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