lipschitz boundary from the polar parametrization

Question

I take $A$ open boundd subset in$\quad \mathbb{R^2}$ and I take the polar parametrization of $A$: $x(\theta)= R(\theta) cos(\theta)$. $y(\theta)= R(\theta)sin(\theta)$.$\quad$ $\theta \in [0,2\pi]$

what is the condition on $R$ to make the boundary of A lipschtizien ?