I am trying to show that the function $θ:\mathbb C\to\mathbb R$ defined by $θ(z)=\arg(z)$ is discontinuous on $(-\infty,0)$.
I am trying to show it first for $-1$, I took two sequences that converges to $-1$ from below$(z_n)$ and from above $(w_n)$.
What I found is $$\lim_{n \rightarrow \infty}{f(z_n)= -\pi},$$ but $$-\pi<\arg(z)\leq \pi,$$so should I say here
$$\lim_{n \rightarrow \infty}{f(w_n)= \pi},$$ and we have
$$\lim_{n \rightarrow \infty}{f(z_n)= \pi},$$ which makes the images for these two sequences have same limits. I AM LOST.
Is there another way to prove that?
Thanks in advance.