I am interesting of finding for all monotone functions $f:[0,1]\to [0,1]$ satisfying the relation $$f\left(2\sum_{i=1}^{\infty}\frac{a_i}{3^i}\right)=\sum_{i=1}^{\infty}\frac{a_i}{2^i}$$ Where $a_i\in \{0,1\}$
I have tired to see whether I could make some conjecture. Namely, such function must satisfy the point-wise relations
$$f(0)=0~~~~~a_i=0~~~~\mbox{for $i\ge 1$}~~~and~~~f(1)=1~~~~~a_i=1~~~~\mbox{for $i\ge 1$}$$
$$f(\frac23)=\frac12~~~~~a_1=1,a_i=0~~~~\mbox{for $i\ge 2$}$$
$$f(\frac13)=\frac12~~~~~a_1=0,a_i=1~~~~\mbox{for $i\ge 2$}$$
continuing this way we can find particular values. But I cannot find a conjecture on this
Can any one help?