Questions about an analytic function from $A$ to $B$

Question

Let $A=\{z\in\mathbb{C} : \vert z\vert \gt1 \}$ and $B=\{z\in\mathbb{C} : z \ne0\}$.

Which of the following are true?

1.There is a continuous onto function $f:A\to B$;

2.There is a continuous one to one function $f:B\to A$;

3.There is a non-constant analytic function $f:A\to B$.

I know that $\mathbb{D}\setminus\{0\} $ is conformally equivalent to $\mathbb{C}\setminus{\overline{D}}$

For 3. $f(z)=\frac{1}{z}$ is non constant analytic function from $A$ to $B$.

How to conclude for other options?

Answer 1

As regards 1. what about $f(z)=z(|z|-1)$?

For 2. try $f(z)=z\left(1+\frac{1}{|z|}\right)$. Does it work?