# Thread: Proving the complex Log function is not continuous

1. ## Proving the complex Log function is not continuous

The question is:

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

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.

2. ## 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 $\displaystyle f(z)=Log(z), z\in \mathbb{C}-(-\infty,0)$
-The point -1, which belongs to the interval $\displaystyle (-\infty,0)$
-Two sequences, $\displaystyle x_n=-1+i/n, y_n=-1-i/n, n\in \mathbb{N}$, which converge to -1.

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

But...