Existence of Nonconstant Periodic Function with no Period

Theorem
There exists a real, non-constant function $f$ such that:


 * $(1): \quad f$ is periodic.


 * $(2): \quad f$ does not have a period.

Proof
By Dirichlet Function is Periodic and Dirichlet Function has no Period, it is seen that the Dirichlet functions are such an example.