At first following is Theorem 8.5 and its proof in Rudin.

[![enter image description here](https://i.stack.imgur.com/LatxD.png)](https://i.stack.imgur.com/LatxD.png)

I don't know why we need the process from "Let A" to "desired conclusion".
Actually without it, Rudin just show that $f(x)=0\ \ \ for \ \ \ |x|<|x-x\_0|+|x\_0|<R$. why we should derive set $A$ is open???