Školjka

%V0 Let $f : [0, 1] \to \mathbb R$ be continuous and satisfy: $$bf(2x) = f(x), \quad 0 \leq x \leq 1/2$$ $$f(x) = b+(1-b)f(2x-1), 1/2 \leq x \leq 1$$ where $\displaystyle b = \frac{1+c}{2+c}$, $c > 0$. Show that $0 < f(x)-x < c$ for every $x, 0 < x < 1.$