Why does the following show Log can't be extended to whole $\mathbb{C}^*$?
Here's another proof which I think I understand, though I'm not sure what's the connection between the two proofs:
I believe the 2nd proof says because approaching from different directions yield different possible values for Log, so Log can't be made continuous. (please tell me if my understanding is wrong)
But one I don't see the connection between the two proofs, and two I don't understand the 1st proof.
I would be happy if anyone could help explain them.