Derivatives and the Carathéodory superposition (Q1175318)

The main result of the paper is the following theorem: If \(D\subset\mathbb{R}^ 2\) is a nonempty open set and \(f:D\to\mathbb{R}\) is a locally bounded function such that all sections \(f^ y\) are derivatives and all sections \(f_ x\) are \(T_ d\)-equicontinuous, then for every continuous and density continuous function \(g:I\to\mathbb{R}\) such that \(I\) is an interval and \((x,g(x))\in D\) for \(x\in I\), the superposition \(h(x)=f(x,g(x))\) is a derivative. Here a family \({\mathcal F}\) of maps of the topological space \((\mathbb{R}\), density topology) into (\(\mathbb{R}\), natural topology) is \(T_ d\)-equicontinuous at \(x\) if for each \(\varepsilon>0\) there is a neighbourhood \(V\in T_ d\) of \(x\) such that \(| f(u)- f(x)|<\varepsilon\) for each \(u\in V\) and \(f\in{\mathcal F}\), and a function \(f:I\to\mathbb{R}\) is density continuous if it is continuous as a map from (\(I\), density topology) into (\(\mathbb{R}\), density topology). The rest of the paper includes two theorems showing the importance of the supposition about sections \(f^ y\).