TITLE: Holomorphic functions and mapping one set to another
QUESTION [5 upvotes]: I was preparing for my exam of complex analysis and i have few questions from previous one that sounded tricky to me.

1) 
    i) Is it possible for $f:\mathbb{C}\to\mathbb{C}$ to be differentiable only in one point?
   
    ii) Analytic only in one point?

My guessing is that first (i) one is true, but cant explain why. Perhaps need an example.
I highly doubt about second (ii) one because for function to be analytic you have to find a domain were function would be differentiable. 

2)  Write down a function
  $f:\mathbb{C}\to\mathbb{C}$ that is
  differentiable in two points only.

I need an example too.
Next question kind'a beat me out. 

3) 
  Make an example of analytic
  $f:\mathbb{C}\to\mathbb{C}$ function
  that maps circle $|z-1| = 1$ to
  circle $|z|=2$ and line $\text{Im } z=0$ to line $\text{Re } z=0$.

My guess is you have to find a fraction function for circle and then use it with composition of $e^{\varphi\theta}$ which rotates the coords. But that's only my guess.

REPLY [3 votes]: For 1i) Consider the function $f(z) = |z|^2$.  It's easy to see that it's not differentiable at any point other than possibly $(0,0)$ by, say, checking the Cauchy-Riemann equations.  A (not so hard) direct calculation shows $f$ really is differentiable at $(0,0)$.