1. ## Function on [0,1]

Hello,

The problem is as follows:

Function $f$ is such that: $f:[0,1] \rightarrow [0,1]$, $f(0)=0$, $f(1)=1$, $f$ is continuous in $0$ and in $1$ and the following condition is satisfied:
$f(x)=2f(x/2)$ for all $x \in (0,1]$.

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

Thanks for help

2. f can not be continuous on neither 0 nor 1.
Do you mean continuous on 0 from right and on 1 from left?

3. Yes, I meant continuous in 0 from right and in 1 from left