Let $\hat{\mathbb{C}} = \mathbb{C} \cup \{ \infty \}$ denote the Riemann Sphere. While reading the wikipedia article about it I found a passage that said that every rational function on the complex plane can be extended to a continuous function on the Riemann Sphere.
The particular construction is as follows, let $R(z) = \frac{f(z)}{g(z)} \in \mathbb{C}(z)$ be a rational function. Lets assume for simplicity that $f(z)$ and $g(z)$ share no common factor. Then for any point $a \in \mathbb{C}$ such that $g(a) = 0$ but $f(a) \neq 0$ we define $R(a) = \infty$. Also we define $R(\infty) := \lim_{z \to \infty} R(z)$.
So to quote wikipedia, with this definitions $R(z)$ becomes a continuous function from the Riemann Sphere to itself. The problem for me is that the article doesn't add any details as to how one may go about showing that in fact $R(z)$ is continuous. So my question is exactly that, to make it simple, if I have for instance $R(z) = \frac{z-1}{z+1}$ or even $R(z) = \frac{1}{z}$, how do I show that it is a continuous function on $\hat{\mathbb{C}}$? I think I can build up from an easy example such as this (assumming this is easy).
Also, I'm a little bit confused about how to interpret this continuity, how should I see the continuity on the Riemann Sphere?, does it involve an argument with stereographic projection?
I added in the Riemann Surface tag just in case they are involved, which I'm not sure. Thank you very much in advance.