ILL 1983 Problem 24

Every $x, 0 \leq x \leq 1$ , admits a unique representation $x = \sum_{j=0}^{\infty} a_j 2^{-j}$ , where all the $a_j$ belong to $\{0, 1\}$ and infinitely many of them are $0$ . If $b(0) = \frac{1+c}{2+c}, b(1) =\frac{1}{2+c},c > 0$ , and $$f(x)=a_0 + \sum_{j=0}^{\infty}b(a_0) \cdots b(a_j) a_{j+1}$$ show that $0 < f(x) -x < c$ for every $x, 0 < x < 1.$