Define $f:\mathbb{\mathbb{C}}^{2}\rightarrow\mathbb{R}$ by
$$f(z_{1},z_{2})=\text{Arg}(z_{1})+\text{Arg}(z_{2})-\text{Arg}(z_{1}z_{2})\;,$$
where $\text{Arg}$ denotes the principal argument. What is the range of the function $f$? (The answer is $\{-2\pi,0,2\pi\}$.)