Question

A function $f: I \to J$ ($I,J$ intervals) has the Darboux property or the Intermediate value property if for every $a < b \in I$ 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.

I know that there are functions $f: [0,1] \to [0,1]$ which have the Darboux property, but have no fixed points.

What are some articles which can be taken as references for this non-existence theorem? The only one I found was this, 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...).

Answer 1

The standard reference for Darboux functions is

Andrew Michael Bruckner and Jack Gary Ceder, Darboux continuity, Jahresbericht der Deutschen Mathematiker-Vereinigung 67 (1965), 93-117. MR 32 #4217l; Zbl 144.30003

http://eudml.org/doc/146526;jsessionid=98522A06CD68A44763F32C1354F068AB

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.

http://www.google.com/search?q=%22Darboux+continuity%22+Bruckner+%22fixed+point%22