Showing that ln z is analytic in $\displaystyle Re \leq 0$ 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 $\displaystyle Re < 0$ Briefly: I showed that $\displaystyle -\pi, \pi $ 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 $\displaystyle Re > 0$ 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.