Membership of Set of Strictly Positive Integers is Replicative Function

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Theorem

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

$\forall x \in \R: f \left({x}\right) = \left[{x \in \Z_{> 0} }\right]$

where $\left[{\cdots}\right]$ is Iverson's convention.


Then $f$ is a replicative function.


Proof

Let $x \in \R$ such that $x > 0$.

Then for all $k \in \Z$ such that $0 \le k < n$:

$x + \dfrac k n \in \Z_{> 0}$

and so from Membership of Set of Integers is Replicative Function:

$\displaystyle \sum_{k \mathop = 0}^{n - 1} \left[{x + \frac k n \in \Z_{> 0} }\right] = \left[{n x \in \Z_{> 0} }\right]$


Let $x \le 0$.

Then for all $k \in \Z$ such that $0 \le k < n$:

$x + \dfrac k n < 1$

and so:

$\displaystyle \sum_{k \mathop = 0}^{n - 1} \left[{x + \frac k n \in \Z_{> 0}}\right] = 0 = \left[{n x \in \Z_{> 0} }\right]$

$\blacksquare$


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