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.