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:

$\forall x \in \R: \map f x = \sqbrk {x \in \Z_{> 0} }$

where $\sqbrk \cdots$ 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:

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


Let $x \le 0$.

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

$x + \dfrac k n < 1$

and so:

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

$\blacksquare$


Sources