Given the knowledge that the exponential function maps bijectively each half-open horizontal strip of width $2\pi$ onto $\mathbb C\setminus \{0\}.$ So, in particular, if we restrict it to $-\pi<\Im z\le \pi$ , we get the principal logarithm Log$:C\setminus \{0\}\to -\pi<\Im z\le \pi$. Then this function is analytic $except$ on the interval $\{-\infty<\Re z\le 0\}.$
I'm trying to see what happens as I approach any given point in the interval $\{-\infty<\Re z\le 0\}$ not equal to zero from $\Im z > 0$ and $\Im z < 0$. But I can't see exactly why it's discontinuous on the interval $\{-\infty<\Re z\le 0\}$.
For any $x$ on the negative real axis write down $log (x+\frac i n)$ and $Log(x-\frac i n)$. It is quite easy to see that these sequences tend to different limits.