Show that $f(x)$ is not continuous at every point $x_0$ not at the origin.
$f(x) = \begin{cases} 2x, & \text{x rational} \\ -2x, & \text{x irrational} \end{cases}$
My working so far:
Using the converse statement,
$\exists\epsilon>0$ such that $\forall\delta>0, \exists x_\delta$ such that $|x_\delta-x_0|<\delta$ and $|f(x_\delta)-f(x_0)|\ge\epsilon$
Then, $|2x_\delta-2x_0|<2\delta??$......and now I'm stuck. How do I go about finding $\epsilon$ and $x_\delta$ ?
I understand there are multiple questions on epsilon-delta proofs but a lot of them simply say choose $\epsilon$ as _ without really explaining. Also the fact that the proof isn't for a specific point is throwing me off.