What can be $f$ so that $f^2(x) = -x$ for all $x\in R$?  I know that if $f^2(x) = -x$ then $f(x)$ is injective and $f$ can not be continuous.  But I can not find an example of discontinuous function so that $f^2(x) = -x$ for all $x\in R$.  Can anyone help me?
I've been perusing the internet looking for interesting problems to solve. I found the following problem and have been going at it for the past 30 minutes with no success:     Find a function $f: \mathbb{R} \to \mathbb{R}$ satisfying $f(f(x)) = -x$ for all $x \in \mathbb{R}$.   I am also wondering, can we find $f$ so that is continuous?  I was thinking of letting $f$ be a periodic function, and adding half the period to x each time. I had no success with this, and am now thinking that such a function does not exist.  Source:
When should I use mmap for file access? POSIX environments provide at least two ways of accessing files.  There's the standard system calls open(), read(), write(), and friends, but there's also the option of using mmap() to map the file into virtual memory.  When is it preferable to use one over the other?  What're their individual advantages that merit including two interfaces? Oven temperature for multiple ducks? Do I need to adjust the oven temperature in a roast duck recipe if I want to put more than one duck in the oven at a time.   I suspect I should keep the same temperature but keep them in longer.  Should I cover each duck in foil individually or together?  Does it matter if they are on the same pan? Biblatex: submitting to the arXiv I am trying to submit a LaTeX file to the  which uses biblatex.  However, it fails whatever I do.    I first get the message   biblatex.sty is not found   I have tried including (all from TeXLive 2010) biblatex.sty but then I get   logreq.sty is missing then logreq.def is missing   then   ! Package biblatex Error: Outdated 'etoolbox' package.   I then tried adding etoolbox.sty and etoolbox.def and get  ! Package biblatex Error: Style 'alphabetic' not found   So I added alphabetic.bbx then standard.bbx then alphabetic.cbx and finally I get   ! Package biblatex Error: Autocite command 'inline' undefined   I also tried incorporating the bib into the main LaTeX file as in  but it doesn't help.  Has anyone successfully used biblatex with the  and if so, how? How do you change the size of figures drawn with Matplotlib? How do you change the size of figure drawn with Matplotlib? Can I kill an indestructible creature with -X/-X and sufficient damage marked on it? Can I hit a 4/4 indestructible creature with a  (3 damage) and give it -1/-1 with  to kill it? (Does it matter which way around I cast those?)  I read  here and it says, damage is removed in cleanup step, so it should be on the creature still after both spells.     702.12. Indestructible   702.12a Indestructible is a static ability.   702.12b A permanent with indestructible can’t be destroyed. Such permanents aren’t destroyed by lethal damage, and they ignore the state-based action that checks for lethal damage (see rule 704.5g).   702.12c Multiple instances of indestructible on the same permanent are redundant. What is this Windows 8 icon for as rectangle icon in the title bar? Every Window in Windows 8 got this icon if it gots the focus. When clicking on it, nothing seems to happen.   How does this icon work? Let  $f:X\times Y\to \mathbb{R}$. I could not come up with an example that satisfies $$\sup_{y\in Y}\inf_{x\in X}f(x,y)&lt;\inf_{x\in X}\sup_{y\in Y}f(x,y).$$ Any help would be appreciated. Thanks! Given $y_n=(1+\frac{1}{n})^{n+1}$ show that $\lbrace y_n \rbrace$ is a decreasing sequence Given  $$ y_n=\left(1+\frac{1}{n}\right)^{n+1}\hspace{-6mm},\qquad n \in \mathbb{N}, \quad n \geq 1. $$ Show that $\lbrace y_n \rbrace$ is a decreasing sequence. Anyone can help ?  I consider the ratio $\frac{y_{n+1}}{y_n}$ but I got stuck. Combinatorial proof for Stirling number of 1st kind's generating function So I see this identity a lot but I can't find a detailed combinatorial proof for it.  $$n(n+1)...(n+k-1)=\sum_{j=0}^k \begin{bmatrix}k \\j \end{bmatrix}n^j$$ Can someone please help? I tried to install ffmpeg in trusty/Ubuntu 14.04 and got the following message:  $sudo apt-get install ffmpeg  Reading package lists... Done Building dependency tree        Reading state information... Done Package ffmpeg is not available, but is referred to by another package. This may mean that the package is missing, has been obsoleted, or is only available from another source  E: Package 'ffmpeg' has no installation candidate   Why isn't the package available? i m using ubuntu 10.10. please tell me how to type in hindi fonts in open office word. please help I need to draw a tree, and constantly using dotty and including the PNGs is starting to be a pain. Is there a way to get LaTeX to draw it's own trees that doesn't involve learning an entirely new language like TikZ? How to evaluate potential advisers on grounds other than their research/publications? I thought it was The Tommyknockers by Stephen King, but after a check with Wikipedia, it appears that's not the case.  What I remember about the story (a TV movie I think, but definitely based on a book) was that the world is created anew every night, and that the old, used up, Earth was eaten nightly by some scary creatures.  The major plot point involved some protagonists not getting onto the new world at midnight, and having to escape from the monsters that were eating up yesterday's planet.  Does this ring a bell for anyone? I'm finding it very hard to Google for. How can I stand on the ground? EM or/and Pauli? There is this famous example about the order difference between gravitational force and EM force. All the gravitational force of Earth is just countered by the electromagnetic force between the electrons on the surface of my feet and the ground.  But also wave-function of electrons(fermions) do not overlap each other due to Pauli exclusion principle.  So which one is the true reason of me not flopping inside the earth forever? Is it only one of them(my guess is Pauli exclusion) or is it both? \numcases with \tag I would like to label some cases with specific tags, and was hoping for something like  \begin{numcases}{A = } \frac{r}{a} &amp; r &gt; a \tag{G} \\ \frac{a}{r} &amp; r &lt; a \tag{H} \end{numcases}   utilizing the numcasesenvironment from the cases package, and \tag from amsmath. However, the two don't seem to be compatible:     ! Package amsmath Error: \tag not allowed here.   Is there any alternatives that would do what I hope to accomplish? Evaluate  $$\lim_{n\to\infty} \left(\frac{1^p+2^p+3^p + \cdots + n^p}{n^p} - \frac{n}{p+1}\right)$$ There's already a question like this  so that my question could be considered duplicate, but I'll try to make my point clear that this is a different question.   Is there a way to derive  from the Lorentz' Force law or just from Maxwell's Equations?   The point is that we usually define, based on experiments, that the force felt by a moving charge on the presence of a magnetic field is $\mathbf {F} = q\mathbf{v}\times \mathbf{B}$, but in that case the magnetic field is usually left to be defined later.   Now can that force law be used in some way to obtain Biot-Savart law like we obtain the equation for the electric field directly from Coulomb's Force law?  I wanted to know that because as pointed out in the question I've mentioned, although Maxwell's Equations can be considered more fundamental, those equations are obtained after we know Coulomb's and Biot-Savart's laws, so if we start with Maxwell's Equations to obtain Biot-Savart's having use it to find Maxwell's Equations then I think we'll fall into a circular argument.   In that case, without recoursing to Maxwell's Equations the only way to obtain Biot-Savart's law is through observations or can it be derived somehow? How do I remove macOS completely from my MacBook Air 2015 and install Windows 10 only after?  I have tried formatting my drive to GPT and tried booting a Windows Installation USB however I'm stuck at the installer not being able to find any drives. Drawing on an image with TikZ