To show that the annulus $\{x \in \Bbb R^2 : 1 < ||x|| < 2\}$ is connected. To show that the annulus $\{x \in \Bbb R^2 : 1 < ||x|| < 2\}$ is connected. 
I want to do it without path-connectedness or polygon-connectedness using the fact continuous image of a connected set is connected.
What will be a suitable function to be defined?

Hint: 
$Q=[0,2\pi]\times (1,2)\subset \mathbb{R}^2$ is connected set and $f:Q\to\mathbb{R}^2$ defined as $f(\theta,r)=(r\cos \theta, r\sin \theta)$ is continuous.
