Used for questions and equations involving the floor function, which is defined to be the function that returns the largest integer less than or equal to $x$ (often denoted by $\lfloor x\rfloor$). See also (ceiling-function).

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18 views

A form for a piecewise continuous function? [on hold]

$\def\rr{\mathbb{R}}$Take any $D \subseteq \rr$. Is it true that for any piecewise continuous function $f : D \to \rr$ there is a continuous function $g : D \to \rr$ and a piecewise constant function $...
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2answers
57 views

Solve in integers the equation $\left\lfloor\frac{x^2-y^3}{x+y^2} \right\rfloor=1+x-y$

Solve in integers the equation $$\left\lfloor\frac{x^2-y^3}{x+y^2} \right\rfloor=1+x-y$$ My attempt: I used http://www.wolframalpha.com/: $x=-2; y=\{3,4,5,6,7,..\}$ or $x=-1, y=\{-10,-9,...\}$. ...
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1answer
31 views

Evaluation of $\bigg[ \lim_{x\to 0} \frac{e-(1+x)^{1/x}}{\tan x}\bigg]$

Find the value of $$\bigg[ \lim_{x\to 0} \frac{e-(1+x)^{1/x}}{\tan x}\bigg]$$ where $[.]$ represents floor function (or greatest integer function) I wrote it as $\bigg[ \lim_{x\to 0} \frac{e-(1+x)^...
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0answers
40 views

Work required to align pieces in a plane.

Given two piecewise continuous functions f(x) and g(x) and that $\lim_{a -> x^-} g(a) - f(a) = \lim_{a -> x^+} g(a) - f(a)$ at all points, find the work used to shift each of the planar slolids ...
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3answers
60 views

How can I remove the floor function from ⌊ab/10⌋?

After several weeks of trying on my own, I was hoping for a hand. I am familiar with transitioning ⌊ab/10⌋ to a mod function as well as a trig function. Ideally, I would like a solution that involved ...
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1answer
29 views

Removing dicontinuity from functions involving modulo?

I am currently looking into removing discontinuity from piecewise continuous functions without changing the derivative where it is defined and (preferably) the value of right sided limit at 0. This is ...
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1answer
80 views

Removing jump discontinuity from a tricky function.

I have the function $\cos(x)\lfloor x \rfloor$ which I would like to make continuous without changing the derivative where it exists or the values approaching 0 from the right side. I can do this by ...
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1answer
84 views

Example of a jump discontinuity where the left and right hand limits do not exist? [closed]

Right off the bat I should probably mention that I am speaking more visually rather than in manners that can be proven rigorously. Please keep that in mind when reading. I'm looking for a function ...
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1answer
62 views

Is the limit $\lim_{x\rightarrow0}\frac{\sin{[x]}}{[x]}$ a one sided limit or not?

Is the limit $\displaystyle\lim_{x\rightarrow0}\frac{\sin{[x]}}{[x]}$ a one sided limit or not? Here $[\, \cdot\, ]$ is the greatest integer function. According to me the right hand limit will be not ...
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2k views

Derivative/integral relationship appears to disprove the fundamental theorem of calculus!!!

Consider the floor function: $$f(x) = \lfloor x \rfloor$$ The indefinite integral of f is: $$\int_0^x f(x) dx = x\lfloor x \rfloor - \frac {\lfloor x \rfloor^2 + \lfloor x \rfloor} 2$$ This should ...
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1answer
75 views

Property of the floor function

Given $u,v \in \mathbb{R}_{+}$, and let $n:=\lfloor v \rfloor$. where $u\in [0,1]$. Is $\lfloor vu \rfloor =\lfloor nu \rfloor$ or $\lfloor vu \rfloor =\lfloor nu \rfloor+1$? Edit: I missed the ...
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2answers
24 views

Limit of floor function

I would like to calculate these limits $\lim\limits_{x \to 0^{+}} \frac{x}{a}{\big\lfloor\frac{b}{x}\big\rfloor}$ $\lim\limits_{x \to 0^{+}} \frac{b}{x}{\big\lfloor\frac{x}{a}\big\rfloor}$, where $a,...
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67 views

Solutions to $\lfloor 2x\rfloor-\lfloor x+1\rfloor=2x$

Find all solutions to $$[2x]-[x+1]=2x$$ where $[x]=\lfloor x\rfloor$ $$$$ I divided this into 2 cases: $$Case 1:x=[x]+\{x\}\text{ where } 0\le\{x\}<0.5$$ $$Case 1:x=[x]+\{x\}\text{ where } 0.5\...
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125 views

Solutions to $\dfrac{1}{\lfloor x\rfloor}+\dfrac{1}{\lfloor 2x\rfloor}=\{x\}+\dfrac{1}{3}$

Find all solutions to $$\dfrac{1}{\lfloor x\rfloor}+\dfrac{1}{\lfloor 2x\rfloor}=\{x\}+\dfrac{1}{3}$$ $$$$ Unfortunately I have no idea as to how to go about this. On rearranging, I got $$3\lfloor ...
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2answers
79 views

Solution of $x-1=(x-\lfloor x \rfloor)(x-\{x\})$

Find all solutions for $$x-1=(x-\lfloor x \rfloor)(x-\{x\})$$ $$$$My approach: $$x-1=\lfloor x \rfloor\{x\}$$ $$\dfrac{x-1}{\lfloor x \rfloor}=\{x\}$$ $$\Rightarrow 0\le \dfrac{x-1}{\lfloor x \...
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1answer
46 views

Series expansion of {x}

Hello and sorry for my bad English. I am not mathematician, so sorry if this seems a silly question. I've seen this formula regarding the fractional part of a number in Wikipedia, and I would like to ...
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3answers
104 views

Solutions to $\lfloor x\rfloor\lfloor y\rfloor=x+y$

Find all solutions to $$\lfloor x\rfloor\lfloor y\rfloor=x+y$$ and show that the non-Integral solutions lie on two unique lines. Also determine the equations of these 2 lines. I divided the problem ...
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1answer
27 views

Summation Closed form for floor$\left(\log_n\right)$

The closed sum for the floors of logs of consecutive integers is: $$\sum_{i=0}^n \lfloor \log_2i\rfloor = n\lfloor \log_2n\rfloor-2^{\lfloor \log_2n\rfloor+1}+\lfloor \log_2n\rfloor+2$$ This formula ...
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2answers
51 views

Integration of floor function.

$\displaystyle{\int_{0}^{3\pi}\left\lfloor\sin\left(x\right)\right\rfloor\,\mathrm{d}x = -\pi \approx -3.14159}$ Here, $\left\lfloor\cdots\right\rfloor$ means floor function Why is it so ?. ...
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1answer
39 views

Prove $\Sigma_{k=0}^{n-1}\lfloor x+\frac{k}{n}\rfloor=\lfloor nx\rfloor$ , n is a Natural Number

Prove the following identity: $$\lfloor x\rfloor +\lfloor x+\frac{1}{n}\rfloor +\lfloor x+\frac{2}{n}\rfloor +\lfloor x+\frac{3}{n}\rfloor+...+\lfloor x+\frac{n-1}{n}\rfloor =\lfloor nx\rfloor$$ ...
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3answers
103 views

Solutions of $\lfloor 4x\rfloor+\lfloor 3x\rfloor=1$

Find all solutions of $$\lfloor 4x\rfloor+\lfloor 3x\rfloor=1$$ I have no idea as to how to go about this question. I would be grateful if somebody would please show me how to solve such questions. ...
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1answer
58 views

A little help with $ x_n = \frac{1}{n^2} \sum_{k=1}^{n} \left[ k \alpha \right] $

So I have this little problem I want to solve that says the following: For every number $\alpha \in \mathbb{R} $, analyze the following sequence $$ \{ x_n \} = \frac{1}{n^2} \sum_{k=1}^{n} \left[ k \...
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2answers
95 views

Analytic floor function, why this seems to work?

I have been using this formula which I determined for myself for quite some time now for use in everything from the sgn() function to the Kronecker delta to the ceiling and NINT() functions but haven'...
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2answers
33 views

Difference of the floor of a product and the product of floors

Is there any way the following can be simplified? $$\lfloor f(x)\cdot g(x) \rfloor - \lfloor f(x) \rfloor \cdot \lfloor g(x) \rfloor$$
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2answers
83 views

Integrating the floor of a function

The integral that I'm trying to simplify is this: (both $x$ and $c$ are natural numbers, if that helps) $$ \mathrm{F}\left(x,c\right) \equiv \int_{0}^{c}\left\lbrace\vphantom{\LARGE A}% \left\lfloor ...
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0answers
36 views

Does $g$ map $\mathbb{R}$ onto the Cantor set?

For $x\in\mathbb{R}$ define \begin{equation} g(x)=1+\tfrac{3}{2}\sum_{k=0}^{\infty}\left(\frac{\left\lfloor2^{2k}x\right\rfloor}{2^{2k}}-\frac{\left\lfloor2^{2k+1}x\right\rfloor}{2^{2k+1}}\right) \...
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1answer
28 views

How does this faulty system of integration change the nature of jump discontinuity?

Let's define a sort of faulty integral. For the purposes of this question we shall assume that this is the regular integral. This integral integrates all functions properly however it's gets confused ...
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1answer
14 views

Removing Discontinuity in 3-space without changing the partial derivative

Is it possible to find a version of the function $$f(x,y) = x\cdot \lfloor y \rfloor + \lfloor x\rfloor^2$$ That is continuous. ANY operation is allowed in changing the function as long as the ...
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1answer
17 views

How to bound a ratio of integer and a real number in floor function?

I have : $$n\le\left\lfloor \dfrac{m}{x}\right\rfloor,$$ where $n$, $m$ are positive integers and $x<1$ is a positive real number. I would like to bound the ratio $\frac{n}{m}$. So I will get: ...
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1answer
19 views

Limit of a floor sum

How can i prove that $ \forall x \in \mathbb{R} \displaystyle \lim_{n \to \infty} \dfrac{\left \lfloor{x}\right \rfloor+\left \lfloor{2x}\right \rfloor+\cdots+\left \lfloor{nx}\right \rfloor}{n^2} = \...
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2answers
30 views

Relation between a floor and a ceiling function for a problem

I was trying to formulate some problem. I want to find a relation between a floor and ceiling function. Suppose the Property 1 satisfies that it has $\lfloor \frac{n}{2} \rfloor$ number of $X$. Then ...
2
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3answers
129 views

Integral: $\int_{0}^{x}\lfloor\dfrac{1}{1-t}\rfloor dt$

I'm working on an integral problem(the rest of which is irrelevant) and this integral arises, which has stumped me. $$\int_{0}^{1}\int_{0}^{x}\left\lfloor\dfrac{1}{1-t}\right\rfloor dt dx$$ $\bf\...
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1answer
34 views

Proving an identity involving floor function

Prove that : $$\left \lfloor \dfrac{2 a^2}{b} \right \rfloor - 2 \left \lfloor \dfrac{a^2}{b} \right \rfloor = \left \lfloor \dfrac{2 (a^2 \bmod b)}{b} \right \rfloor $$ Where $a$ and $...
2
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2answers
23 views

$ \lim_{x\rightarrow a} f(x)= \lim_{x\rightarrow a} [f(x)]$ then at $x=a$ is there a maxima or minima?

$$ \lim_{x\rightarrow a} f(x)= \lim_{x\rightarrow a} [f(x)]$$ Where [.] denotes the greatest integer function (floor) function. $f(x)$ is non-constant in the neighborhood of 'a' and is continuous ...
2
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1answer
34 views

Limit of Series with differences of Floor function

Problem: Evaluate $$ L=\displaystyle \lim_{n \to \infty} \frac{1}{n} \sum_{k=1}^{n} \left( \lfloor \frac{2n}{k} \rfloor -2\lfloor \frac{n}{k} \rfloor \right)$$. Please help me with this one. I ...
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2answers
56 views

Possible alternative for finding the Area under the floor function (aka, the integral of floor(x))

So, I had to ask myself the question as to what the area under the floor function could possibly be. I started by graphing the basic $\mbox{floor}(x)$ function (I personally use desmos.com for a nice ...
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2answers
182 views

What is $\lfloor i\rfloor$?

So, floor is a function that converts a real number to an integer. It rounds down. This makes sense; however, what about complex numbers? I know that depending on the number, it can be split linearly. ...
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36 views

Inequality with floored squareroots

$(\lfloor \sqrt{n}\rfloor +1)^2\ge n+1$, for all $n\in \mathbb{N}$. I have convinced myself that this is true, but would like to see a formal proof.
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1answer
63 views

Area enclosed by the curve $\lfloor |x''| \rfloor +\lfloor |y''| \rfloor = 2$

The area enclosed by the curve $$\bigg\lfloor \frac{|x-1|}{|y-1|}\bigg\rfloor +\bigg\lfloor \frac{|y-1|}{|x-1|}\bigg\rfloor = 2\;,$$ Where $-2 \leq x,y\leq 0$ $\bf{My\; Try::}$ Let $x-1=x'$ and $y-1 ...
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2answers
45 views

When does $\lfloor\sqrt{x^2 + n}\rfloor = \lfloor\sqrt{(x-1)^2 + n}\rfloor$?

Call $x \in[\sqrt{n},n/2] \cap \Bbb{Z} $ a critical point if the following holds $$\lfloor\sqrt{x^2 + n}\rfloor = \lfloor\sqrt{(x-1)^2 + n}\rfloor$$ It appears that mostly this does not happen. For ...
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1answer
40 views

Solve $\lfloor x \rfloor = ax+1$ for integral $a$

Solve $\lfloor x \rfloor = ax+1$, where $a$ is an integer. I have found the values of $x$ for $a=0$, $a=1$ and $a=-1$. But I don't know how to continue. How can I find the solutions for integral $a\...
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2answers
39 views

Summation over a floor function of a first degree polynomial

I've been trying to solve a difficult programming question for the last four days. I've gotten most of it done, but the piece I can't seem to figure out is this: Find a closed form expression of $$\...
4
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3answers
51 views

Sum of number-of-divisors function equals $\sum_{j=1}^{n} \lfloor n/j \rfloor$.

I am trying to prove the identity $$t(1) + t(2) + \cdots + t(n) = \Big\lfloor \dfrac{n}{1} \Big\rfloor + \Big\lfloor \dfrac{n}{2} \Big\rfloor + \cdots + \Big\lfloor \dfrac{n}{n} \Big\rfloor,$$ where ...
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1answer
21 views

Proof of $\sum_{j=1}^{p-1} \lfloor jq/p \rfloor = \frac{1}{2}(q-1)(p-1)$ Involving Pairing of Summands

I've seen the proof of the identity $$\sum_{j=1}^{p-1} \lfloor jq/p \rfloor = \frac{1}{2}(q-1)(p-1)$$ where $p$ and $q$ are coprime positive integers. This involves counting the remainders $r_{j}/p$...
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3answers
49 views

Moving terms outside of a floor

If I have $x^{\lfloor\frac{c}{a-b}\rfloor}$ is this equivalent to $(x^{\lfloor\frac{1}{a-b}\rfloor})^{\lfloor c \rfloor}$ if a,b,c are all integers and x is between 0 and 1? I'm concerned with this ...
2
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0answers
18 views

How to prove $|n-m|\lt d\implies |f(n)-f(m)|\lt e$ for the following condition?

How to prove $\exists n\in\Bbb{N},\forall e\in \Bbb{R^+},\exists d\in\Bbb{R^+},\forall m\in\Bbb{N}, |n-m|\lt d\implies |f(n)-f(m)|\lt e$ if $f(x)=\lfloor \frac{n}{3}\rfloor$. I don't know how to ...
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5answers
60 views

How to prove or disprove $\forall x\in\Bbb{R}, \forall n\in\Bbb{N},n\gt 0\implies \lfloor\frac{\lfloor x\rfloor}{n}\rfloor=\lfloor\frac{x}{n}\rfloor$.

How to prove or disprove $\forall x\in\Bbb{R}, \forall n\in\Bbb{N},n\gt 0\implies \left\lfloor\frac{\lfloor x\rfloor}{n}\right\rfloor=\left\lfloor\frac{x}{n}\right\rfloor$. So we want to prove $\left\...
1
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1answer
77 views

Different ways of evaluating $n!$?

I've recently managed to prove the following result and was hoping to know if it already exists? $ \def\lf{\left\lfloor} \def\rf{\right\rfloor}$ $$ \ln(n!) = \sum_{k=1}^{p_k < n}\left( \sum_{r=1}...
2
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0answers
139 views

Find polynomial congruent to function modulo 6

How to find polynomial with rational coefficients, which is congruent to given function (or prove that it doesn't exist)? Particularly, how to find $P(x)$, so $$P(x)\equiv 4\left\lfloor\frac{x}{6}\...
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1answer
56 views

Sums involving floor function

I am looking for a direct formula for this sum $$\sum_{k=0}^n \lfloor{\sqrt{n+k}}\rfloor\lfloor{\sqrt{k}}\rfloor$$ Or a method to efficiently compute the sum for large n