Školjka

%V0 For all rational $x$ satisfying $0 \leq x < 1$, f is defined by $$f(x)=\left\{\begin{array}{cc}\frac{f(2x)}{4},&\mbox{ for }0\leq x <\frac{1}{2},\\ \frac{3}{4}+\frac{f(2x-1)}{4}, &\mbox{for }\frac{1}{2}\leq x < 1.\end{array}\right$$ Given that $x = 0.b_1b_2b_3 \cdots$ is the binary representation of $x$, find, with proof, $f(x).$