$\Omega\subset\mathbb{R}^2$ defined in polar coordinates by $-\pi<\theta<\pi$ and $r>1$. Prove $u=\theta$ is not Lipschitz continuous in $\Omega$

Question

Let $\Omega\subset\mathbb{R}^2$ be the open set defined in polar coordinates by $-\pi<\theta<\pi$ and $r>1$. I have to prove that the function $u=\theta$ is not Lipschitz continuous in $\Omega$. How can I explicitly see this is true? I tried using the definition of Lipschitz continuous but I can't go anywhere from there.

Answer 1

Hint: Consider $u(-2+ih) - u(-2-ih)$ as $h\to 0^+.$