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 φ :[0,1]→ [0,1] φ : [ 0 , 1 ] → [ 0 , 1 ] , distinct from the identity on [0, 1], such that φ (0)=0 φ ( 0 ) = 0 , φ (1)=1 φ ( 1 ) = 1 and φ (x)=φ (x/2)+φ (x+1/2)-φ (1/2) φ ( x ) = φ ( x 2 ) + φ ( x + 1 2 ) - φ ( 1 2 ) for every x∈ [0,1] x ∈ [ 0 , 1 ] ? By now, it is known that each of the de Rham functions Rₚ R p , where p∈ (0,1) p ∈ ( 0 , 1 ) , is a solution of the MW-problem, and for any Borel probability measure μ μ concentrated on (0, 1) the formula φ _μ (x)=∫ (0,1)Rₚ(x)\, dμ (p) ϕ μ ( x ) = ∫ ( 0 , 1 ) R p ( x ) d μ ( p ) defines a solution φ _μ :[0,1]→ [0,1] 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 μ .