I have proved a few years ago the following proposition:

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 $A,B\neq \emptyset$)

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?