Articles with examples of Darboux functions without fixed points - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-23T09:17:21Z http://mathoverflow.net/feeds/question/93636 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/93636/articles-with-examples-of-darboux-functions-without-fixed-points Articles with examples of Darboux functions without fixed points Beni Bogosel 2012-04-10T09:16:44Z 2012-04-12T19:11:02Z <p>A function $f: I \to J$ ($I,J$ intervals) has the Darboux property or the Intermediate value property if for every $a &lt; b \in I$ and for every $\lambda$ between $f(a)$ and $f(b)$ there exists $c \in [a,b]$ such that $f(c)=\lambda$. Equivalently, the image of any interval under $f$ is an interval.</p> <p>I know that there are functions $f: [0,1] \to [0,1]$ which have the Darboux property, but have no fixed points. </p> <blockquote> <p>What are some articles which can be taken as references for this non-existence theorem? The only one I found was <a href="http://projecteuclid.org/DPubS?verb=Display&amp;version=1.0&amp;service=UI&amp;handle=euclid.rae/1184963801&amp;page=record" rel="nofollow">this</a>, but I guess that there are older articles which deal with this subject. I searched Google and Mathscinet, but didn't find any except the one above (maybe I don't know how to search...).</p> </blockquote> http://mathoverflow.net/questions/93636/articles-with-examples-of-darboux-functions-without-fixed-points/93690#93690 Answer by Dave L Renfro for Articles with examples of Darboux functions without fixed points Dave L Renfro 2012-04-10T17:37:38Z 2012-04-10T17:37:38Z <p>The standard reference for Darboux functions is</p> <p>Andrew Michael Bruckner and Jack Gary Ceder, <em>Darboux continuity</em>, Jahresbericht der Deutschen Mathematiker-Vereinigung 67 (1965), 93-117. MR 32 #4217l; Zbl 144.30003</p> <p><a href="http://eudml.org/doc/146526;jsessionid=98522A06CD68A44763F32C1354F068AB" rel="nofollow">http://eudml.org/doc/146526;jsessionid=98522A06CD68A44763F32C1354F068AB</a></p> <p>Given its publication date, I don't think you'll find much about fixed points of Darboux functions in this paper, but you should look over the paper anyway, plus it's an excellent survey that should be quite useful in general. However, and this is a useful search tip, you can use the paper to refine your search. Search google using the phrases "Darboux continuity" AND "Bruckner" AND "fixed point". The idea is that the hits you get will be those papers and other items that cite Bruckner/Ceder's paper, and thus you'll exclude the more superficially researched items.</p> <p><a href="http://www.google.com/search?q=%22Darboux+continuity%22+Bruckner+%22fixed+point%22" rel="nofollow">http://www.google.com/search?q=%22Darboux+continuity%22+Bruckner+%22fixed+point%22</a></p>