Lift on non star-shaped domain

Question

I came across the following exercise.

Let $$ A = \{ x \in \mathbb{R}^2 : 1 \leq \lvert x \rvert \leq 2 \}. $$ Show that there is no continuous function $\vartheta: A \rightarrow \mathbb{R}$ such that

$$ E(x) := \frac{x}{\lvert x \rvert} = (\cos\vartheta(x), \sin \vartheta(x)) \quad \text{for all} \; x \in A. \quad \quad (*) $$

I know that $(*)$ would hold, if $A$ was star-shaped and $E$ a continuous unit vector field.

However, I am not able to find a contradiction.

Answer 1

Hint: Note $e^{it} = e^{i\vartheta (e^{it})}.$ Thus $\vartheta (e^{it}) = t +2\pi n_t,$ where $n_t\in \mathbb Z.$