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arXiv · 2203.16333

On some nested floor functions and their jump discontinuities

Luca Onnis
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Abstract

This paper investigates some particular limits involving nested floor functions. We'll prove some cases and then we'll show a more general result. Then we'll count the discontinuity points of those functions, and we'll prove a method to find them all. Surprisingly the set of the jump discontinuities of $f_n$ is a subset of the set of the jump discontinuities of $f_{n+1}$, $\forall n\in\mathbb{Z^{+}}$ where: \[ f_n(x)=\underbrace{\Biggl\lfloor x\Bigl\lfloor x \lfloor\dots\rfloor\Bigr\rfloor\Biggr\rfloor}_{\text{$n$ times}} \] Furthermore we'll give some generalizations of the result and lots of considerations; for example we'll prove that the cardinality of the set of the discontinuities of $f_n$ in a given limited interval approaches infinity as $n\to\infty$.

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Luca Onnis. 2022-03-28. On some nested floor functions and their jump discontinuities. https://arxiv.org/abs/2203.16333

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