Let $\hat{\mathbb{C}}$ be the riemann sphere, and let $U\subseteq \hat{\mathbb{C}}$ be an open set. We say a function $f:U\to \hat{\mathbb{C}}$ with $f(\mathrm{\infty })\ne \mathrm{\infty }$ is differentiable at $\mathrm{\infty }$ if 1;;$g(z)\underset{def}{=}f\left(\frac{1}{z}\right)$ is differentiable at $0$.

Suppose that instead we have that $f(z_0)=\mathrm{\infty }$ for $z_0\ne \mathrm{\infty }$. Then we say that $f$ is differentiable at $z_0$ if ==1;;$g(z)=\frac{1}{f(z)}$ is differentiable at $z_0$