Wikipedia on Bertrand's 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 deviations are as
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?