## Contest problem

There exist functions $f:[0,1]\to [0,1]$ with Darboux property, such that there exist sets $A,B$ with $A\cap B=\emptyset,\ A\cup B=[0,1]$ and $f(A)\subseteq B,\ f(B)\subseteq A$.
Problem proposed in the “Traian Lalescu” contest for undergraduate students in Romania in 2004.

This problem can be solved using some ideas from Sierpinski’s Theorem. The proof can be found here.