# Subtract Half is Replicative Function

## Theorem

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

$\forall x \in \R: f \left({x}\right) = x - \dfrac 1 2$

Then $f$ is a replicative function.

## Proof

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

Hence the result by definition of replicative function.

$\blacksquare$