# General Periodicity Property/Corollary

## 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:

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

$\blacksquare$