# Questions tagged [floor-function]

The floor function, also known as the greatest integer function, maps a real number $x$ to the greatest integer less than or equal to $x$ (often denoted $\lfloor x \rfloor$). See also (ceiling-function).

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### Greatest Integer Function linear equation

Given that $2[x]=x+2(x)$, $[x]$ if the Greatest Integer Function and $(x)$ is the fractional part of $x$, find the value (s) of $x$. I tried replacing $(x)=x–[x]$ but for an equation in $x$ and $[x]$....
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### Evaluate $\int_{1}^{n} \lfloor x \rfloor^{x- \lfloor x \rfloor} dx$.

$\int_{1}^{n} [x]^{x-[x]} dx$ I tried to approach this with riemann sum method but it seems impossible by that way. Even using other general integration techniques it seems quite complicated .I have ...
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### How do limits work with floor/ceiling?

I'm interested in the below equation: $$\frac{n}{\operatorname{floor}(\frac{x}{n})}$$ Plotting with $n = 1..100$ shows the graph being slightly more aliased as $n$ increases and a discontinuity ...
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### Reasoning about inequalities involving floor functions

I am working on the beginning of an inductive argument and I wanted to confirm that my base case is sound. Let $f(x) = \lfloor x\rfloor - \left\lfloor\dfrac{x}{2}\right\rfloor$ where is $x$ is a ...
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### Showing $\int_{a}^{b} \left\lfloor x \right\rfloor dx + \int_{a}^{b} \left\lfloor -x \right\rfloor dx=a-b$

I want to show $$\int_{a}^{b} \left\lfloor x \right\rfloor dx + \int_{a}^{b} \left\lfloor -x \right\rfloor dx=a-b$$ I know that \left\lfloor -x \right\rfloor = \begin{cases} -\...
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### Prove/disprove that $\lfloor x\rfloor \leq t \iff x\leq\lfloor t\rfloor +1$
Prove/disprove that $\lfloor x\rfloor \leq t \iff x\leq\lfloor t\rfloor +1$ Playing around I can see why this is true, but I have no idea how to prove that, any ideas?