Subtract Half is Replicative Function

Theorem

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

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

$\ds \sum_{k \mathop = 0}^{n - 1} \map f {x + \frac k n}$

$=$

$\ds \sum_{k \mathop = 0}^{n - 1} \paren {x - \frac 1 2 + \frac k n}$

$\ds $

$=$

$\ds n x - \frac n 2 + \frac 1 n \sum_{k \mathop = 0}^{n - 1} k$

$\ds $

$=$

$\ds n x - \frac n 2 + \frac 1 n \frac {n \paren {n - 1} } 2$

$\ds $

$=$

$\ds n x - \frac n 2 + \frac n 2 - \frac 1 2$

$\ds $

$=$

$\ds n x - \frac 1 2$

$\ds $

$=$

$\ds \map f {n x}$

Hence the result by definition of replicative function.

$\blacksquare$