Subtract Half is Replicative Function

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Theorem

Let $f: \R \to \R$ be the real function defined as:

$\forall x \in \R: \map f x = x - \dfrac 1 2$


Then $f$ is a replicative function.


Proof

\(\displaystyle \sum_{k \mathop = 0}^{n - 1} \map f {x + \frac k n}\) \(=\) \(\displaystyle \sum_{k \mathop = 0}^{n - 1} \paren {x - \frac 1 2 + \frac k n}\)
\(\displaystyle \) \(=\) \(\displaystyle n x - \frac n 2 + \frac 1 n \sum_{k \mathop = 0}^{n - 1} k\)
\(\displaystyle \) \(=\) \(\displaystyle n x - \frac n 2 + \frac 1 n \frac {n \paren {n - 1} } 2\) Closed Form for Triangular Numbers
\(\displaystyle \) \(=\) \(\displaystyle n x - \frac n 2 + \frac n 2 - \frac 1 2\)
\(\displaystyle \) \(=\) \(\displaystyle n x - \frac 1 2\)
\(\displaystyle \) \(=\) \(\displaystyle \map f {n x}\)

Hence the result by definition of replicative function.

$\blacksquare$


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