Let $\operatorname{Arg}: \Bbb{C} \setminus \{0\} \to\Bbb{R}$ be the principal value of the argument, taking values in $(−\pi, \pi]$. Using the standard metrics on $\Bbb{C} \setminus \{0\}$ and $\Bbb{R}$, show that $\operatorname{Arg}$ is not continuous.
I am thinking about using the standard metric $d_1(z,w) = |z-w|$ and showing that there does not exist a $\delta$ with $|z-w|<\delta$ for all $\epsilon >0$ and all $z$ in $\Bbb{C}$.
Let $f(z)= Arg (z)$ for $z \ne 0.$ And let $x_0 \in \mathbb R$ with $x_0<0.$
Compute $ \lim_{n \to \infty}f(x_0+i \frac{1}{n})$ and $ \lim_{n \to \infty}f(x_0- i \frac{1}{n})$.
Conclusion ?
No. You have the quantifiers wrong.
What you need is to show that there exists a $\varepsilon>0$ such that for any $\delta>0$, there exist $z,w\in\mathbf{C}\setminus\{0\}$ such that $$ |z-w|<\delta,\quad\textrm{and}\quad |\operatorname{Arg}(z)-\operatorname{Arg}(w)|>\varepsilon. $$
Now you can look for a point $z_0$ that has the argument $\pi$, which are the only discontinuities of the argument function. Consider, for instance, $z_0=-1$ and two points $z,w$ near it. Draw a picture to guide your intuition.
For the formal details, you may set $z=e^{i(\pi-\theta)}$ and $w=e^{i(-\pi+\theta)}$ for small $\theta>0$.