General Periodicity Property/Corollary

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Theorem

Let $f: \R \to \R$ be a real function.


Then $L$ is a periodic element of $f$ if and only if:

$\forall x \in \R: \map f {x \bmod L} = \map f x$

where $x \bmod L$ is the modulo operation.


Proof

Necessary Condition

Let $f: \R \to \R$ be a real function with a periodic element $L$.

Then:

\(\displaystyle \map f x\) \(=\) \(\displaystyle \map f {n L + r} \quad 0 < r < \size L\) Definition of Quotient
\(\displaystyle \) \(=\) \(\displaystyle \map f r\) General Periodicity Property
\(\displaystyle \) \(=\) \(\displaystyle \map f {x \bmod L}\)

$\Box$


Sufficient Condition

Let $f: \R \to \R$ be a real function such that for all $x \in \R$:

$\map f {x \bmod L} = \map f x$


Let $n = \floor {\dfrac x L}$.

Then:

\(\displaystyle \map f {x + L}\) \(=\) \(\displaystyle \map f {\paren {x + L} \bmod L}\)
\(\displaystyle \) \(=\) \(\displaystyle \map f {x + L - \floor {\frac {x + L} L} L}\)
\(\displaystyle \) \(=\) \(\displaystyle \map f {x + L - \paren {n + 1} L}\)
\(\displaystyle \) \(=\) \(\displaystyle \map f {x - n L}\)
\(\displaystyle \) \(=\) \(\displaystyle \map f {x \bmod L}\)
\(\displaystyle \) \(=\) \(\displaystyle \map f x\)

$\blacksquare$