Hello,

The problem is as follows:

Function $\displaystyle f$ is such that: $\displaystyle f:[0,1] \rightarrow [0,1]$, $\displaystyle f(0)=0$, $\displaystyle f(1)=1$, $\displaystyle f$ is continuous in $\displaystyle 0$ and in $\displaystyle 1$ and the following condition is satisfied:

$\displaystyle f(x)=2f(x/2)$ for all $\displaystyle x \in (0,1]$.

The question is: Does only function $\displaystyle f(x)=x$ satisfy the conditions above? We may prove that if $\displaystyle f$ is either convex or concave then this is the only solution. But can we do the same without assuming that?

Thanks for help