Singular Strictly Increasing Functions and a Problem on Partitions of Closed Intervals

We establish that the problem of constructing a strictly increasing singular function is equivalent to the problem of constructing subsets $\mathscr P$ and $\mathscr Q$ of a closed interval $[a;b]\subset\mathbb R$ such that (1) $\mathscr P\cap\mathscr Q=\varnothing$; (2) $\mathscr P\cup\mathscr Q=[a;b]$; (3) the Lebesgue measures of the intersections of $\mathscr P$ and $\mathscr Q$ with an arbitrary interval $J\subset[a;b]$ are positive.