How do I proof that changing the angle between the $X$ and $Y$ axis of a cartesian plane will move each point on the plane continuously?

Question

I'm doing some work on different coordinate systems and I need help proofing something:

Let's say I have a cartesian plane and draw lines parallel to the $x$ axis through each point $y = 1,2,3,-1,-2,-3$ etc., then I change the angle between each of these lines, including the $x$ axis, by $\alpha = 30$ such that all the points are moved and still reside on the lines.
Here is an image showing the transformation I'm trying to do:

I need to prove that each point is moves in an continuous way, i.e. the route taken by each point can be drawn without taking your pen of the paper (I'm sorry if this term isn't the right one).

Is this doable?