论文标题
在某些嵌套地板功能及其跳跃不连续性
On some nested floor functions and their jump discontinuities
论文作者
论文摘要
本文研究了一些涉及嵌套地板功能的特定限制。我们将证明某些情况，然后将显示一个更普遍的结果。然后，我们将计算这些功能的不连续点，我们将是一种找到所有功能的方法。出乎意料的是，$ f_n $的跳跃不连续性集是$ f_ {n+1} $，$ \ forall n \ in \ mathbb {z^{+}} $ were：wery：： \ [ f_n（x）= \ UnderBrace {\ biggl \ lfloor x \ bigl \ bigl \ lfloor x \ lfloor \ dots \ rfloor \ rfloor \ bigr \ bigr \ rfloor \ biggr \ biggr \ rfloor} _ { \] 此外，我们将对结果和许多考虑进行一些概括。例如，我们将证明在给定的有限间隔接近$ f_n $的不连续性的基数将无穷大在$ n \ to \ infty $中。
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 $n\to\infty$.
