It turns out I misunderstood the exercise somewhat. I had to show that ln z is analytic in the whole complex plane, but the negative real part. This expression got the better of me. \
Showing that ln z is analytic in I used CR-equations. I guess that's sufficient since a function needs to be analytic to satisfy C-R equations.
I'm also supposed to show that ln z is discontinuous in Briefly: I showed that hits the same point if the radius is the same.
Now to my question, not a part of the exercise, but I like to know. How can I show that ln z is continuous in for instance I have another thread about continuous functions, but that's a more general prof. I like to put get together an example with some help from you folks.