I'm trying to construct a well-defined $C^\infty$ function $f(x,y) : \mathbb{R}^2 \setminus \{ (x,0) \mid x \leq 0 \} \to (-\pi, \pi)$ via polar coordinates, which cannot be continuously extended to $\mathbb{R}^2 \setminus \{ (0,0) \}$.
However, I'm not sure how to even start coming up with ideas for how $f$ should look. I thought that perhaps something involving $\arctan$ might be useful, but I'm not really sure that this counts as using polar coordinates. Also, for example if I set $f(x,y) = 2\arctan(y)$, then I only hit $(\pi, \pi) \setminus \{ (0,0) \}$; I'm not sure how to also hit $(0,0)$.