In basic calculus, one partial-differentiate a differentiable function whose domain is an open set or a closed set etc. However how formally this process works?
Here is a reference : definition of winding number, have doubt in definition.
Define $p:\mathbb{C}\rightarrow \mathbb{C}\setminus\{0\}:z\mapsto e^z$.
Then, it is a covering map, hence if $\alpha:[0,1]\rightarrow \mathbb{C}\setminus\{0\}$ is a given path, there is a lift $\tilde{\alpha}$ such that $p\circ \tilde{\alpha}=\alpha$, by Homotopy lifting theorem. This means that $\alpha$ can be decomposed into continuous "length" part and "angle" part.
However, consider a continuous function $f:V\rightarrow \mathbb{C}\setminus\{0\}$ where $V$ is open in $\mathbb{C}$. Saying $f$ can be decomposed into continuous length part and angle part, means that there exists a lift $g$ of $f$. Does this always exist? If so how? If not, why is polar coordiate introduced so uncautiously?
Not necessarily, for example if $V=\mathbb C\setminus\{0\}$ itself and $f$ is the identity. Then the argument part can't be chosen continuously.
It should be true if $V$ is required to be simply connected rather than open.