Möbius
transformations are continuous on the Riemann
sphere
The
complex plane together with the point at infinity ( ) make up the Riemann sphere.
The North pole of the Riemann sphere corresponds to , the South pole
corresponds to the origin of the complex plane and the equator corresponds
to the unit circle.
In order to show that Möbius transformations are continuous
on the Riemann sphere, we must look at two cases:
1) If c = 0,
and
2) If c ≠ 0,
then and
Therefore, M(z) is continuous on the extended z plane, or Riemann sphere.
