Here is my problem:

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?

Thank you for hints and help.