On the metric Darboux property
Let \(f\in L_ p(0,1)\) and \[ \omega_ p(f;h):=\sup_{0<t\leq h}\{\int^{1-t}_{0}| f(x+t)-f(x)|^ pdx\}^{1/p} \] be it's modulus of continuity. A function \(f\in S(0,1)\) is named D-function, if for any decomposition an interval \(I\subset(0,1)\) into two measurable sets \(E_ 1\), \(E_ 2\), \(mes E_ i>0,\) one has \( \sup_{E_ 1}f\geq \inf_{E_ 2}f.\) The main result is the following Theorem. If \(\underline{\lim}_{h\to 0}h^{1/p}\omega_ p(f;h)<\infty,\) then \(f=f_ 0+f_ 1,\) where \(f_ 0\) is a D-function, and \(f_ 1\) has left- side and right-side limits at any point of \([0,1].\) The result cannot be improved in terms of the modulus of continuity \(\omega_ p(f;.)\).
scientific article; zbMATH DE number 3865618
scientific article