On some nested floor functions and their jump discontinuities

TL;DR

This paper studies nested floor functions, proves new results about their limits and discontinuities, and introduces a method to identify all jump discontinuities, revealing their increasing complexity with higher nesting levels.

Contribution

It provides a general framework for analyzing nested floor functions, characterizes their discontinuities, and shows how the set of discontinuities grows with nesting depth.

Findings

Discontinuity points of $f_n$ are a subset of those of $f_{n+1}$.

The number of discontinuities in a fixed interval tends to infinity as $n$ increases.

A method to find all jump discontinuities of nested floor functions.

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 $n\to\infty$.