Existence of a continuous one to one map from the set $A= \{z \in \Bbb C : z \neq 0 \}$ to the set $B = \{z \in \Bbb C : |z|>1 \}$.

Question

Does there exist a continuous one to one map from the set $A= \{z \in \Bbb C : z \neq 0 \}$ to the set $B = \{z \in \Bbb C : |z|>1 \}$?

I have tried to find this map though I have failed to do this. Please help me in this regard.

Thank you very much.

Answer 1

$$f(z)=e^{\lvert z\rvert}\frac{z}{\lvert z\rvert}.$$

Answer 2

$f(z)=z+\frac{z}{|z|}.$

The geometric meaning is: for a vector $z\neq 0$, we put it away from the origin $1$ unit in the direction of $z$, the inverse is pull it back $1$ unit.