  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
$\mu$.
