Do Meromorphic Functions automatically give us Entire Functions?

Question

Suppose I have a meromorphic function, $f$. Then, I can write $f(z)=\frac{h(z)}{g(z)}$ where $h,g$ are entire. I would really like to be able to claim that $f$ extends to an entire function by analytic continuation (is this from Riemann?), but is that allowed here? I always thought this was a given, but I suppose now I can't quite wrap my head around it. Would it be because $f$ is bounded in each deleted neighborhood of $g$? Anyway, I would appreciate any help! Thank you.

Answer 1

If, say, $h(z)=1$ and $g(z)=z$, then $h$ and $g$ are entire. However, $f(z)=\frac{g(z)}{h(z)}=\frac1z$, which is meromorphic, cannot be extended to an entire function, since $\lim_{z\to0}\frac1z=\infty$.