SCHWEITZER 2016 Problem 5

Does there exist a piecewise linear continuous function $f:\mathbb{R}\to \mathbb{R}$ such that for any two-way infinite sequence $a_n\in[0,1]$ , $n\in\mathbb{Z}$ , there exists an $x\in\mathbb{R}$ with $$\limsup_{K\to \infty} \frac{\#\{k\le K\,: k\in\mathbb{N},f^k(x)\in[n,n+1)\}}{K}=a_n$$ for all $n\in\mathbb{Z}$ , where $f^k=f\circ f\circ \dots\circ f$ stands for the $k$ -fold iterate of $f$ ?