Monotone sections of functions of two variables (Q1821890)

The paper deals with the following question: Let P be a property of a real function defined on \(<0,1>\), let \(f:<0,1>\times <0,1>\to R\), let \(f_ x:<0,1>\to R\) \((f^ y:<0,1>\to R)\) be a section of f for x(y), i.e. \(f_ x(y)=f(x,y)\) for every \(y\in <0,1>\) \((f^ y(x)=f(x,y)\) for every \(x\in <0,1>)\). Let \(A_ 1(f)=\{x\in <0,1>:f_ x\) has the property P\} and \(A_ 2(f)=\{x\in <0,1>:f^ y\) has the property P\}. For the properties: ''nondecreasing'', ''increasing'', ''nondecreasing and continuous'', ''increasing and continuous'' and ''of bounded variation'', the author gives necessary and sufficient conditions for the couple \((A_ 1,A_ 2)\) of subsets of \(<0,1>\) for the existence of such a function \(f:<0,1>\times <0,1>\to R\) for which \(A_ 1=A_ 1(f)\) and \(A_ 2=A_ 2(f)\).