Here is my problem:

There is given the functional equation: $\displaystyle f(2x)+f(1/2)=f(x)+f(x+1/2)$ for $\displaystyle x \in [0,1/2]$. We also know that $\displaystyle f(0)=-1$ and $\displaystyle f(1)=1$. Additionally, we assume the continuity and strict monotonicity of $\displaystyle f$. Is it possible to get any information on $\displaystyle f(1/2)$? In particular, can we prove that $\displaystyle f(1/2)=0$?

Thank you for hints and help.