A problem about polar coordinates of $\mathbb{R}^2$

Question

In Bott-Tu's book, differential forms in Algebraic Topology, they used the angle function $\theta$ in $\mathbb{R}^2-\{0\}$, and its differential $d\theta$, how to understand them, since $\theta$ is not continuous in $\mathbb{R}^2-\{0\}$.