Exploring Bertrand's Theorem: Why is β the Same Rational Number for All Radii?

[Kashmir]

Wikipedia on Bertrands theorem, when discussing the deviations from a circular orbit says:
>..."The next step is to consider the equation for $u$ under small perturbations ${\displaystyle \eta \equiv u-u_{0}}$ from perfectly circular orbits"

(Here $u$ is related to the radial distance as $u=1/r$ and $u_0$ corresponds to the radius of a circular orbit ) ...>"The solutions are
${\displaystyle \eta (\theta )=h_{1}\cos(\beta \theta )}$">"For the orbits to be closed, $β$ must be a rational number. What's more, **it must be the same rational number for all radii**, since β cannot change continuously; the rational numbers are totally disconnected from one another"Why does $\beta$ have to be the **same** rational number for all radii at which a circular orbit is possible ?

I understand why it should be rational, but why the same number for all radii?

Link: https://en.m.wikipedia.org/wiki/Bertrand's_theorem

 

[A.T.]

"For the orbits to be closed, $β$ must be a rational number. What's more, **it must be the same rational number for all radii**, since β cannot change continuously; the rational numbers are totally disconnected from one another"

Why does $\beta$ have to be the **same** rational number for all radii at which a circular orbit is possible ?

As the quote states, rational numbers are totally disconnected from one another, meaning that between two rational numbers you have non-rational numbers. So the rational number β cannot change continuously, while you continuously vary the radius.