Question about proof of Liouville's theorem

Question

I am trying to work through a proof of Liouville's theorem here and ran into an issue. In his answer (as I understand it) Eremenko first establishes that $|f(0)|\leq \max_{|z|=r}|f(z)|$ for entire functions and proposes to apply this to the function $g(z) = (f(z)-f(0))/z$ to get $$|g(0)|\leq \frac{\max_{|z|=r}|f(z)| + |f(0)|}{r}$$ and then take the limit as $r\to \infty$ to obtain that $|f'(0)|=\lim_{z\to 0} |g(z)|$ is zero. But how do we know this function $g$ is entire? Do we not have a problem at zero?

Answer 1

The fact that $\lim_{z \to 0} g(z)$ exists (and equals $f'(0)$) implies that $g$ has removable singularity at $0$. Defining $g(0)$ to be $f'(0)$ makes $g$ an entire function.