# Proving the complex Log function is not continuous

• Apr 25th 2013, 05:20 AM
CuriosityCabinet
Proving the complex Log function is not continuous
The question is:

Let $Log z = ln \left | z \right | + i \theta$ where $-\pi < \theta \leq \pi$and $z=\left | z \right | e^{i\theta}$, $(z\neq 0)$. Prove that Log is not continuous on $(- \infty,0).$

Hint: Consider the sequences {-1+i/n} and {-1-i/n}.

I am not sure how I should be using these sequences in the proof.

• Oct 16th 2016, 04:28 AM
Rebesques
Re: Proving the complex Log function is not continuous
There's something alluring about giving late replies.

Let's see what we have.
-The function $f(z)=Log(z), z\in \mathbb{C}-(-\infty,0)$
-The point -1, which belongs to the interval $(-\infty,0)$
-Two sequences, $x_n=-1+i/n, y_n=-1-i/n, n\in \mathbb{N}$, which converge to -1.

If the function were continuous at -1, then the limits $\lim f(x_n), \lim f(y_n)$ should be equal.

But...