An argument function $\phi$ on $\mathbb{C}\setminus\{0\} = \mathbb{R}^2\setminus\{0\}$ is a function such that for every $z\neq 0$ it holds that $z = |z|\exp(i\phi(z)).$
Is there an elementary and easy proof that there is no continuous argument function on $\mathbb{C}\setminus\{0\}$? I would like to see a proof which uses as less complex analysis as possible. Probably only topological arguments and no complex numbers whatoever?