General Antiperiodicity Property

Theorem
Let $f: X \to X$ be an anti-periodic function, where $X$ is either $\R$ or $\C$.

Let $L$ be an anti-periodic element of $f$.

Let $n \in \Z$ be an integer.


 * If $n$ is even, then $nL$ is a periodic element of $f$.
 * If $n$ is odd, then $nL$ is an anti-periodic element of $f$.

Proof
Suppose that $X = \C$.

Case 1
If $n$ is even, then:

Case 2
If $n$ is odd, then:

The proof for when $X = \R$ is nearly identical.