Mobius Transformations, quick question concepts.

[binbagsss]

So a mobius transformation is defined as \frac{az+b}{cz+d}=f(z).
Where ad-bc≠0.


My question is just deriving this condition ad-bc≠0.

I understand that the condition describes the case were the mapping leaves all points unchanged. This is described as undefined in some textbooks...(why is this, isn't it then just an identity map?)

But not by setting z=f(z).

I can't see why this condition would not equally be desribed by z=f(z)?

Many Thanks for any assistance.

 

[maajdl]

If ad-bc=0 then f(z) = b/d which is a constant.
Therefore, the equation y=f(z) can be inverted with respect to z only if ad-bc is not zero.

 

[binbagsss]

But I can't see why this condition would not equally be desribed by z=f(z)?

 

[Dick]

But I can't see why this condition would not equally be desribed by z=f(z)?

I guess I don't see why anyone would describe that mapping where ad-bc=0 as 'leaving all points unchanged'. It doesn't. It maps everything into a constant.