NMC 2000 Problem 4

The real-valued function $f$ is defined for $0 \leq x \leq 1, f(0)=0, f(1)=1$, and

$$\frac{1}{2} \leq \frac{f(z)-f(y)}{f(y)-f(x)} \leq 2$$

for all $0 \leq x<y<z \leq 1$ with $z-y=y-x$. Prove that

$$\frac{1}{7} \leq f\left(\frac{1}{3}\right) \leq \frac{4}{7}$$