Question 11
(FIN 2′) Let $f:[0,1]\to\mathbb R$ be continuous and satisfy:
$$\begin{aligned}bf(2x)&=f(x),&&0\le x\le1/2;\\f(x)&=b+(1-b)f(2x-1),&&1/2\le x\le1,\end{aligned}$$
where $b=\frac{1+c}{2+c}$, $c>0$. Show that $0\lt f(x)-x\lt c$ for every $x$, $0\lt x\lt 1$.
$$\begin{aligned}bf(2x)&=f(x),&&0\le x\le1/2;\\f(x)&=b+(1-b)f(2x-1),&&1/2\le x\le1,\end{aligned}$$
where $b=\frac{1+c}{2+c}$, $c>0$. Show that $0\lt f(x)-x\lt c$ for every $x$, $0\lt x\lt 1$.