Dirichlet Function has no Period

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Theorem

The Dirichlet functions are periodic by Dirichlet Function is Periodic.

However, they do not admit a period.


That is, there does not exist a smallest value $L \in \R_{\gt 0}$ such that:

$\forall x \in \R: \map D x = \map D {x + L}$


Proof

Let $D: \R \to \R$ be a Dirichlet function.


In proving that the Dirichlet Function is Periodic, it was shown that every non-zero rational number is a periodic element of $D$.

Therefore, the period of $D$ must be the smallest element of $\Q_{\gt 0}$.

But from Rational Numbers are not Well-Ordered under Conventional Ordering there cannot exist such an element.

Hence the result.

$\blacksquare$