Proof of Liouville's theorem - simple question

[stgermaine]

Homework Statement

i'm using Fisher's Complex Variables for my complex analysis class and there is a proof for Liouville's theorem. It says "Set g(z)=(F(z) - F(0)) / z; Then g is an entire function"

Homework Equations

N/A

The Attempt at a Solution

I am confused by that statement. Doesn't g have a singularity at z=0? How can it be analytic over the entire complex plane?

Thank you and sorry if it seems like a very stupid question.

 

[Jufro]

I don't have that particular complex book. So, I am not sure about that particular proof. But do you think that for the simple reason that as z tends to 0 you will end up with the indiscriminate form 0/0 has something to do with it?