A function $f:[a,b] \to [a,b]$ is said to have a fixed point $c \in [a,b]$ if $f(c)=c$. Show that every continuous function $f$ mapping $[a,b]$ onto itself has at least one fixed point. I came up with a basic proof of this. I don't think it's quite correct, but let me know what you think.  Suppose $f:[a,b] \to [a,b]$ is continuous on $[a,b]$ Since the domain and codomain of $f$ are the same, $f(x)=x$ Let $\varepsilon&gt;0, c \in[a,b]$ Let $\delta= \varepsilon$ Suppose $|x-c|&lt;\delta$ $\Rightarrow |x-c|&lt;\varepsilon$ $\Rightarrow |f(x)-f(c)|&lt;\varepsilon$ Thus, $\lim_{x \to c} f(x)=f(c)$ and therefore $f(x)$ is continuous for all $c \in [a,b]$ Then $f(c)=c$ for all $c \in [a,b]$ Thus, every continuous $f$ mapping $[a,b]$ onto itself has at least one fixed point
Question: Let $a, b \in \mathbb{R}$ with $a &lt; b$ and let $f: [a,b] \rightarrow [a,b]$ continuous. Show: $f$ has a fixed point, that is, there is an $x \in [a,b]$ with $f(x)=x$.  I suppose this has to do with the basic definition of continuity. The definition I am using is that $f$ is continuous at $a$ if $\displaystyle \lim_{x \to a} f(x)$ exists and if $\displaystyle \lim_{x \to a} f(x) = f(a)$. I must not be understanding it, since I am not sure how to begin showing this... Should I be trying to show that $x$ is both greater than or equal to and less than or equal to $\displaystyle \lim_{x \to a} f(x)$ ?
Sending form POST request to PHP VSE isn't multi-threading Whenever I render a video using the blender video editor only one core is ever used at any time. I have an 8 core CPU(FX-9590) yet regardless of whether blender is set to auto detect or a fixed 8 threads only one core ever sits at 100% at any given time. The rest of the cores sit and really low usage as if blender is completely ignoring them. I'm running blender 2.77a on Linux. I'm using Theora as the codec with an ogg container and Vorbis audio. How do I get blender to use all my cores? Alternatively, why does the force created by blowing out air feel so much stronger than the force created by sucking in air?  Ok, so forget the human factor involved in blowing out candles. Consider a vacuum cleaner with a suction end and a blower end. Anyone who has tried it out notices the blower end creates a much stronger force than the suction end, despite the discharge being (more or less) equal at both ends.  Why does this happen? Before Voldemort, who was the last Horcrux creator? Horcruxes are rare (and then seven come along at once...) and only a few people know about them.  But when was the last Horcrux before Voldemort's? 50 years before him? Centuries? Millennia? I have been having extreme difficulties with this integral. I would appreciate any and all help. $$ \int \sqrt{\tan x} ~ \mathrm{d}{x}. $$ How do i know which wheels will fit my bike? my bike is:  and everything I have came with the bike, only now the axle of the rear wheel bent and the skewer has lost its grooves.  and I would like to replace the rear wheel with the 12 x 142mm version of:   I want to reuse the old casettes and disc break. I was told to buy new rotor bolts, will these do?   Will I be able to migrate all my previous parts onto this new wheel and will the wheel fit onto my bike?  update: A replacement axle would be prefered. What axle would work for my bike? Undefined control sequence BibTex This is one of the BibTex entries giving the problem:  @ARTICLE{2015ApJ...808L..36B,    author = {{Bonnivard}, V. and {Combet}, C. and {Maurin}, D. and {Geringer-Sameth}, A. and      {Koushiappas}, S.~M. and {Walker}, M.~G. and {Mateo}, M. and      {Olszewski}, E.~W. and {Bailey}, III, J.~I.},     title = "{Dark Matter Annihilation and Decay Profiles for the Reticulum II Dwarf Spheroidal Galaxy}",   journal = {\apjl}, archivePrefix = "arXiv",    eprint = {1504.03309},  primaryClass = "astro-ph.HE",  keywords = {dark matter, galaxies: dwarf, galaxies: individual: Reticulum II, gamma rays: galaxies, methods: statistical, stars: kinematics and dynamics},      year = 2015,     month = aug,    volume = 808,       eid = {L36},     pages = {L36},       doi = {10.1088/2041-8205/808/2/L36},    adsurl = {http://adsabs.harvard.edu/abs/2015ApJ...808L..36B},   adsnote = {Provided by the SAO/NASA Astrophysics Data System} }   and corresponding bbl file giving the error at compilation is  \bibitem{2009ApJ...697.1071A} {\sc W.~B. {Atwood}, A.~A. {Abdo}, M.~{Ackermann}, W.~{Althouse},   B.~{Anderson}, M.~{Axelsson}, L.~{Baldini}, J.~{Ballet}, D.~L. {Band},   G.~{Barbiellini}, and et~al.} \newblock {\bf {The Large Area Telescope on the Fermi Gamma-Ray Space Telescope   Mission}}. \newblock {\em \apj}, {\bf 697}:1071--1102, June 2009.   My guess is that Kile doesn't like references of the form 2015ApJ...808L..36B since these are the only ones giving the error.    Can somebody hint me to the problem in this? Thank you very much How do I determine which graphics card I'm using? bilinear form - proof I have to prove that the mapping $f(x,y)={\displaystyle \sum_{i=1}^{n}}{\displaystyle \sum_{j=1}^{n}}x_{i}y_{j}{f}(e_{i},e_{j})$  is a bilinear form, that is, inter alia, the condition: $f(x+y,z)=f(x,z)+f(y,z)$  I have: $f(x+y,z)=f\left({\displaystyle \sum_{i=1}^{n}}x_{i}e_{i}+{\displaystyle \sum_{k=1}^{n}}y_{k}e_{k},{\displaystyle \sum_{j=1}^{n}}z_{j}e_{j}\right)=f\left({\displaystyle \sum_{i=1}^{n}}{\displaystyle \sum_{k=1}^{n}}\left(x_{i}e_{i}+y_{k}e_{k}\right),{\displaystyle \sum_{j=1}^{n}}z_{j}e_{j}\right)=\ldots$  and what's next?  Help! Vitali-type set with given outer measure Remove entire row in a file if first column is repeated What's the difference between  var A = function () {     this.x = function () {         //do something     }; };   and  var A = function () { }; A.prototype.x = function () {     //do something }; Background reflected in glossy glass, but I didn't set it I've made a roughness 0 glass pane, when I'm in lookdev mode it's reflecting a background that I didn't put there. I know I could just ignore it, but I'd rather be able to see what it'll mostly look like when I'm in the later stages. I also just want to know how on Earth I managed to set this background without realising it  Reflection completely ignores any objects I put in front of it. If I placed this glass inside my house, it would still just reflect trees (This is the point where I felt I couldn't ignore it - reflections are going to play a big part in my project and I don't want to be in render mode all the time)  The nodes are the default glass BSDF, except I set roughness to 0  The glass is set to 20mm thick, if that makes a difference  The background moves with my perspective and moves when I relocate the glass  I have a few HDRI's on my PC, but none look remotely like this   Also, I'm using cycles as I find Evee buggy as hell I'm trying to split a string up into words and punctuation, adding the punctuation to the list produced by the split.  For instance:  &gt;&gt;&gt; c = "help, me" &gt;&gt;&gt; print c.split() ['help,', 'me']   What I really want the list to look like is:  ['help', ',', 'me']   So, I want the string split at whitespace with the punctuation split from the words.  I've tried to parse the string first and then run the split:  &gt;&gt;&gt; for character in c: ...     if character in ".,;!?": ...             outputCharacter = " %s" % character ...     else: ...             outputCharacter = character ...     separatedPunctuation += outputCharacter &gt;&gt;&gt; print separatedPunctuation help , me &gt;&gt;&gt; print separatedPunctuation.split() ['help', ',', 'me']   This produces the result I want, but is painfully slow on large files.  Is there a way to do this more efficiently? Use spray paint to make mask for pre-sensitised board? I'm about to etch my first simple board for a project. I have a pre-sensitised board that (as I understand the process) you expose to UV light, then develop the board in a sodium hydroxide solution, then etch. The UV light will negatively affect the protective lacquer and developing it after exposing washes away the lacquer, making the board ready to etch.  I got the lye and I got sodium persulfate for the etching but I did not have any transparent paper as I thought I did! And it's impossible for me to get a hold of it in time (= this weekend, I live far from the city and it's a small one and I haven't found any place that stocks it so I have to order online and that's sometime next week I'd get it at the fastest).   So I am looking for an alternative solution. What I am considering is printing the design to regular paper, cut out a mask carefully from this paper and place it over the board and paint a mask on. Spray paint most likely, then in theory I would have a UV mask.  Anyone see any flaws in this plan? Should flagging a question for closure prevent close-voting? Can White Walkers talk? Through all the five seasons of Game of Thrones we had never seen a White Walker talking.  Does this mean they can't talk human language (e.g. they have their own language) or we will never see them talking (i.e. they can't talk at all)? As a new user on a site, what's the best way to ensure your question will be positively received? Why am I getting 50V at a light when the switch is off? How to make grass appear on top of ground not in middle?