Suppose a continuous function $f:[0,1] \to \mathbb{R}$ satisfies the following equation for all $z \in \left(0,\frac{1}{2}\right)$, $$\int_z^{2z} [f(x)-f(z)] dx = 0.$$ It is clear that a constant function $f(x)=c$ satisfies it. I would like to prove that there are no other such continous functions.

Note: this is a missing part of a larger proof I'm working on. I've already verified that if $f$ is a polynomial then it must be constant. Any hints on how to prove it for arbitrary continuous functions would be appreciated.