At what points does the function $f(z)=Arg(z)$ , $ z \in \mathbb Z-\{0\}$ have a limit? At what points does the function $f(z)=Arg(z)$, $z \in \mathbb Z-\{0\} $ have a limit ?
I know that $-\pi<Arg(z) \leq\pi$. Then how can I use that property to find points where limit exists ?
A where is $f(z)$ is continuous ?

Note that,
$$
\mathrm{Arg}(z)=\mathrm{Im}\,\log z.
$$
The function $\log z$ is definable in every simply connected domain $\Omega$ which does not contain $z=0$. Hence, $\mathrm{Arg}(z)$ is continuous (and indeed real analytic) in every such $\Omega$.
Conversely, if $\mathrm{Arg}(z)$ is defined, as a continuous function, in a simply connected domain $\Omega$ which does not contain $z=0$, then $\log z$ defined as an analytic function.
