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:
- $\ds \sum_{k \mathop = 0}^{n - 1} \sqbrk {x + \frac k n \in \Z_{> 0} } = \sqbrk {n x \in \Z_{> 0} }$
Let $x \le 0$.
Then for all $k \in \Z$ such that $0 \le k < n$:
- $x + \dfrac k n < 1$
and so:
- $\ds \sum_{k \mathop = 0}^{n - 1} \sqbrk {x + \frac k n \in \Z_{> 0} } = 0 = \sqbrk {n x \in \Z_{> 0} }$
$\blacksquare$
Sources
- 1997: Donald E. Knuth: The Art of Computer Programming: Volume 1: Fundamental Algorithms (3rd ed.) ... (previous) ... (next): $\S 1.2.4$: Integer Functions and Elementary Number Theory: Exercise $39 \ \text{c)}$