Over the years, I've often heard that there is no logarithm function which is continuous on $\mathbb{C}\setminus\{0\}$.
The usual explanation is usually some handwavey argument about following such a function around the unit circle, and getting a contradiction at $e^{2\pi i}$ or something.
I've been a little unsatisfied with these. What is a more formal, rigorous proof that there is no continuous log function on $\mathbb{C}\setminus\{0\}$ that is understandable to a nonexpert? Thanks.