# Are there periodic functions without a smallest period?

The Wikipedia page for periodic functions states that the smallest positive period $P$ of a function is called the fundamental period of the function (if it exists). I was intrigued by the condition that the function actually has a smallest period, so my question is, what properties of a function would cause it to be periodic but not have a smallest period?

For a nontrivial example, consider the Dirichlet function, which has $$\delta(x) = \begin{cases}0 & \text{ if x is rational}\\1 & \text{ if x is irrational}\end{cases}$$
Then $\delta(x)$ is periodic with period $r$ for every rational number $r$.