Analytic function in extended complex plane

Question

If we define an extended complex plane by using stereographic projection (Riemann's sphere) $\bar{\mathbb{C}}=\mathbb{C}\cup\infty$, it makes a function $f(z)=\begin{cases}\frac{1}{z}, z\neq 0\\ \infty, z=0 \end{cases}, f:\mathbb{C}\rightarrow \bar{\mathbb{C}}$ continuous ($z=0$ is removable discontinuity point for $\frac{1}{z}$), but what about the analytic property of the function? Do meromorphic functions ever become analytic (or even whole), if we add $\infty$ to the codomain? If we add $\infty$ to both domain and codomain, we get that only meromorphis functions are rational, meaning the set of analytic functions isthe subset of rational functions. What happens if we add it only to codomain?

Answer 1

The analytic functions from $\mathbb C$ to $\overline{\mathbb C}$ correspond to the meromorphic functions on $\mathbb C$, i.e. a pole $p$ of $f$ is an isolated singularity of $f$ such that $|f(z)| \to \infty$ as $z \to p$.

The analytic functions from $\overline{\mathbb C}$ to $\mathbb C$ are constants