Functional equation

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