## Proof That Extended Mobius Transformations Preserve Regions and Boundaries

Theorem

Let be an extended mobius transformation and let be a region in the domain of (not including the point infinity). Then is a region and hat f maps the boundary of to the boundary of in Proof is analytic and non constant on so it follows from the Open Mapping Theorem that is a region.

To show that maps the boundary in of onto the boundary in of we show that if is any open disc in centred at where then there is an open disc centred at such that There are three cases:

a) If is in the domain of and then is continuous at so for each disc there is a disc such that b) If and is a pole of then as so for each disc centred at there is such that whenever Moreover so whenever Thus if then c) If then has a removable singularity or pole at so for a disc centred at there is such that whenever but this means that whenever and so if is the disc then Next suppose that is a boundary point in of and let be any open disc centred at then there is an open disc centred at such that But is a boundary point in of and so contains points from both and It follows that and hence contains points from both and so is a boundary point in of To show the mapping is onto we apply the same argument to and show that if is a boundary point in of then is a boundary point in of The result follows. 