Another look at the Matkowski and Wesołowski problem yielding a new class of solutions

The following MW-problem was posed independently by Janusz Matkowski and Jacek Wesołowski in different forms in 1985 and 2009, respectively: Are there increasing and continuous functions \(\varphi :[0,1]\rightarrow [0,1]\) $\phi :[0,1]\to [0,1]$ , distinct from the identity on [0, 1], such that \(\varphi (0)=0\) $\phi (0)=0$ , \(\varphi (1)=1\) $\phi (1)=1$ and \(\varphi (x)=\varphi (\frac{x}{2})+\varphi (\frac{x+1}{2})-\varphi (\frac{1}{2})\) $\phi (x)=\phi (\frac{x}{2})+\phi (\frac{x+1}{2})-\phi (\frac{1}{2})$ for every \(x\in [0,1]\) $x\in [0,1]$ ? By now, it is known that each of the de Rham functions \(R_p\) $R_p$ , where \(p\in (0,1)\) $p\in (0,1)$ , is a solution of the MW-problem, and for any Borel probability measure \(\mu \) $\mu$ concentrated on (0, 1) the formula \(\phi _\mu (x)=\int _{(0,1)}R_p(x)\, d\mu (p)\) ${\varphi }_{\mu }(x)={\int }_{(0,1)}{R}_p(x)d\mu (p)$ defines a solution \(\phi _\mu :[0,1]\rightarrow [0,1]\) ${\varphi }_{\mu }:[0,1]\to [0,1]$ 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 \(\mu \) $\mu$ .