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

The following MW--problem was posed independently by Janusz Matkowski and Jacek Weso{}owski in different forms in 1985 and 2009, respectively: Are there increasing and continuous functions φ [0,1]→ [0,1], distinct from the identity on $[0,1]$, such that φ(0)=0, φ(1)=1 and φ(x)=φ(x/2)+φ(x+1/2)-φ(1/2) for every x∈[0,1]? By now, it is known that each of the de Rham functions Rₚ, where p∈(0,1), is a solution of the MW--problem, and for any Borel probability measure μ concentrated on $(0,1)$ the formula φ_μ(x)=∫(0,1)Rₚ(x) dμ(p) defines a solution φ_μ[0,1]→[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 μ.