Is sign function continuous in complex plane?

Question

Is sign function continuous in the complex plane? Is it continuous everywhere except at $z = 0$ ? Is it continuous in a circle of unit radius and argument having $\exp(-i\theta)$ ?

Answer 1

$\frac{|z|}{z}$ is continuous in $\mathbb{C} \setminus \{0\}$ as it is the quotient of continuous functions. However, $\frac{|z|}{z}$ is obviously not defined in $z = 0$ (hence not continuous there either).

By the way, it is more common to use $\frac{z}{|z|}$ instead of $\frac{|z|}{z}$ because then you have $z = |z| \cdot \frac{z}{|z|}$ just like for the reals.