Smooth extension of the angle function $\theta(x,y)$

Question

Consider the vector field \begin{align} F:\ \mathbb R^2\setminus\{(0,0)\}&\longrightarrow\mathbb R^2 \\ (x,y)&\longmapsto F(x,y)=\left(\frac{-y}{x^2+y^2},\frac{x}{x^2+y^2}\right) \end{align} We know that $F$ has a potential function $$\theta(x,y)=\arctan\frac{y}{x},\ x\ne0$$ which is also known as $\textbf{angle function}$. I wonder if $\theta$ could be extended to a smooth function on $\mathbb R^2\setminus\{(0,0)\}$. So where should I start ? Could anyone hint me some solid idea ? Thanks