Is punctured disc a Lipschitz domain

Question

I don't quite understand how to apply the definition (Understanding Lipschitz domain) of Lipschitz domain. My question is about annulus (of which punctured disc is a special case). Is annulus a Lipschitz domain. How can we write its boundary(two disjoint circles) as the graph of a Lipschitz map locally? I know we can cover these circles by charts locally.

Answer 1

The annulus, i.e., $$ A_{r,R}=\{(x,y):r^2<x^2+y^2<R^2\} $$ where $0<r<R$ is a Lipschitz domain, since its boundary is the level set of a Lipschitz continuous function.

However, for $R>r=0$, the punctured disc $$ A=\{(x,y):0<x^2+y^2<R^2\} $$ is not a Lipschitz domain.