query: Surjective function: $x^2- \lfloor x \rfloor^2$?
positive_passages.docid: 146023
positive_passages.text: How to prove that $x^2 - \lfloor x\rfloor^2$ is onto?
negative_passages.docid: 27844
negative_passages.text: How do we prove that $\lfloor0.999\cdots\rfloor = \lfloor 1 \rfloor$?
negative_passages.docid: 19064
negative_passages.text: Proof about floor function: $\lfloor x\rfloor+\lfloor x + \frac{1}{n} \rfloor+\cdots + \lfloor x + \frac{(n-1)}{n} \rfloor = \lfloor nx \rfloor$
negative_passages.docid: 158395
negative_passages.text: Find $\lfloor {\alpha}^6 \rfloor$
negative_passages.docid: 136842
negative_passages.text: $|x-f(x)|<5777$, Show that $f$ is surjective
negative_passages.docid: 99483
negative_passages.text: If $T(n+1)=T(n)+\lfloor \sqrt{n+1}, \rfloor$ $\forall n\geq 1$, what is $T(m^2)$?
negative_passages.docid: 5934
negative_passages.text: How to prove floor identities?
negative_passages.docid: 64583
negative_passages.text: Number of surjective functions from A to B
negative_passages.docid: 5222
negative_passages.text: $\lfloor x\rfloor + \lfloor y\rfloor \leq \lfloor x+y\rfloor$ for every pair of numbers of $x$ and $y$
negative_passages.docid: 30639
negative_passages.text: Efficient computation of $\sum_{k=1}^n \left\lfloor \frac{n}{k}\right\rfloor$
negative_passages.docid: 124847
negative_passages.text: Is the set of surjective functions from $\mathbb{N}$ to $\mathbb{N}$ uncountable?
negative_passages.docid: 157334
negative_passages.text: A valid floor function trick?
negative_passages.docid: 91759
negative_passages.text: Evaluate $\lim_{x \to 0} (x\lfloor\frac{1}{x}\rfloor)$
negative_passages.docid: 40973
negative_passages.text: How to find $\lfloor 1/\sqrt{1}+1/\sqrt{2}+\dots+1/\sqrt{100}\rfloor $ without a calculator?
negative_passages.docid: 47923
negative_passages.text: For $n\in\mathbb{N}$, evaluate $\int_0^n \cos(2\pi \lfloor x\rfloor\{x\})dx$
negative_passages.docid: 146306
negative_passages.text: A continuous surjective function from $(0,1]$ onto $[0,1]$
negative_passages.docid: 17285
negative_passages.text: Proof of greatest integer theorem: floor function is well-defined