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.

$\blacksquare$