I have proved a few years ago the following proposition:



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> There exists $f: [0,1] \to [0,1]$ with Darboux property such that there exist $A,B \subset[0,1]$ with $A\cap B=\emptyset,\ A \cup B=[0,1]$ with $f(A)\subset B$ and $f(B)\subset A$. (of course $A,B\neq \emptyset$)
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A function $f : I\subset \Bbb{R} \to \Bbb{R}$ ($I$ is an interval) has the Darboux property if $f([a,b])$ is an interval forall $[a,b]\subset I$.


The proof resembles the proof of Sierpinski's Therem, that any function $f : \Bbb{R} \to \Bbb{R}$ can be written as the sum of two functions each of them having the Darboux property.


My question is:


* have I proved something new, or it is a known fact that such a function exists?
* if the proposition is original can it be useful, I mean, can I submit this as an article?


[edit:] I know I should have done my job and send this to a some magazines to see if it is worth publishing. One of my teachers said that I send the article to JMAA, and of course it got rejected, because it's not that good. I tried at another magazine, but didn't even get an answer if it is rejected or not. I thought then that the proposition is not worthy of an article and proposed it as a problem to AMM. They said its too hard to be published as a problem. As I am a beginner and don't have any paper published until now. I don't know where should I try to send it. 


Could you please name some magazines where I could try and send the paper and recieve an answer to wether the proposition can be published or not?



Darboux function on $[0,1]$ with interesting property
Just a comment, and a hint. The property of the function $f$ that you consider, that is, *The interval $[0,1]$ admits a partition in two sets that are mapped into each other by $f$,* may be restated equivalently as: *No iterate of $f$ of odd order has a fixed point (or also, $f$ has no periodic point of odd period*). The equivalence is quite obvious, but this puts your result into the category: *Darboux property vs Fixed point property,* which seems to have a certain interest, looking on Google. A journal that recently had some papers on this topic, and seems suitable for your result, is [*Real Analysis Exchange*](http://www.stolaf.edu/analysis/). Also, you may consider if your method extends to the construction of a Darboux function with no periodic points at all.


