If $G$ is an open connected subset of $\mathbb{C}$ that does not contain the origin, we call a continuous function $\alpha$ satisfying $\alpha(z) = \text{arg} z$ for all $z \in G$ a branch of arg $z$.
Prove there is no branch of arg z (and consequently no continuous complex logarithm) in the region $D^* =0 < |z| < 1$.
Attempt at a proof:
Assume there is a branch of arg $z$, i.e., there exists $\alpha : D^* \to [a,b] \subset \mathbb{R}$ such that $\alpha(z) = \text{arg} z$. Consider the path $\gamma : [0,1] \to D^*$ defined by $\gamma(t) = \frac{1}{2} e^{2 \pi i t}$.
Then, clearly $\gamma(0) = \gamma(1)$.
Moreover, the function $t \mapsto \alpha(\gamma(t)) = \text{arg}(\gamma(t))$ is a strictly increasing function. Hence, $\alpha(\gamma(0)) < \alpha(\gamma(1))$ contradicting the fact that $\gamma(0) = \gamma(1)$.
My concern with the outlined proof is that I don't know how to actually show that $t \mapsto \alpha(\gamma(t))$ is a strictly increasing function... other than the fact that intuitively this is so. Is it sufficient to say that arg($e^z$) = Im($z$)?
Thanks! Alternative proof styles are also welcome!
