Another look at the Matkowski and Weso{\l}owski problem yielding a new class of solutions

The following MW--problem was posed independently by Janusz Matkowski and Jacek Weso{\l}owski in different forms in 1985 and 2009, respectively: Are there increasing and continuous functions $\varphi\colon [0,1]\to [0,1]$, distinct from the identity on $[0,1]$, such that $\varphi(0)=0$, $\varphi(1)=1$ and $\varphi(x)=\varphi(\frac{x}{2})+\varphi(\frac{x+1}{2})-\varphi(\frac{1}{2})$ for every $x\in[0,1]$? By now, it is known that each of the de Rham functions $R_p$, where $p\in(0,1)$, is a solution of the MW--problem, and for any Borel probability measure $\mu$ concentrated on $(0,1)$ the formula $\phi_\mu(x)=\int_{(0,1)}R_p(x) d\mu(p)$ defines a solution $\phi_\mu\colon[0,1]\to[0,1]$ of this problem as well. In this paper, we give a new family of solutions of the MW--problem consisting of Cantor-type functions. We also prove that there are strictly increasing solutions of the MW--problem that are not of the above integral form with any Borel probability measure $\mu$.