So I have this one confusion on determining continuity of the argument of some complex number $w$. The example goes, on the annulus $\alpha = \{w: 1<|w|<2\},$ $\arg w$ cannot be defined continuously.
My Questions: What does it even mean for the argument of the complex number $w$, to be continuous? If so could someone please explain why this $\arg$ for this particular region cannot be defined continuously?
Thanks in advance.
What it means is that on this annulus $A$, there is no continuous function $f:A\to \Bbb R$ with the property that $$w=|w|\exp(i f(w)).$$
To see this consider the function $\phi:[0,2\pi]\to A$ defined by $\phi(t)=(3/2)\exp(it)$. One must have $f(\phi(t))=t+2\pi n_t$ where $n_t$ is an integer, possibly depending on $t$. But then $n_t$ must be independent of $t$, by continuity. Thus $f(\phi(t))=t+2\pi n$. In particular $f(3/2)=f(\phi(0))=2\pi n$ and $f(3/2)=f(\phi(2\pi))= 2\pi+2\pi n$, a contradiction.