Baire classification of generalized extreme derivatives (Q1063709)

Let \(x\in R=(-\infty,\infty)\), a path leading to x is a subset \(E_ x\) of R such that \(x\in E_ x\) and x is a point of accumulation of \(E_ x\). A system of paths is an indexed system \({\mathcal E}=\{E_ x: x\in R\},\) where \(E_ x\) is a path leading to x. Let \({\mathcal K}\) be the space of all compact non-empty subsets of R with the Hausdorff metric. Then the system \({\mathcal E}=\{E_ x: x\in R\}\) of paths is called continuous iff for each x \(E_ x\) is a compact subset of R and the function \(\hat E: R\to {\mathcal K}\) defied by \(\hat E(\)x)\(=E_ x\) for each \(x\in R\) is continuous. Let \(f: R\to R\) and \({\mathcal E}=\{E_ x: x\in R\}\) be a system of paths, then \(f'_{{\mathcal E}}(x)=\lim_{y\to x,y\in E_ x- \{x\}}\frac{f(y)-f(x)}{y-x}\) when this limit exists and \(\bar f'_{{\mathcal E}}(x)= \limsup_{y\to x,y\in E_ x-\{x\}}\frac{f(y)-f(x)}{y-x}.\) The main results are: Let \(f: <0,1>\to R\) be continuous and \({\mathcal E}=\{E_ x: x\in <0,1>\}\) be a continuous system of paths, then 1) If \(f'_{{\mathcal E}}(x)\) exists for each \(x\in <0,1>\), then \(f'_{{\mathcal E}}\) is of the Baire class one; 2) \(\bar f_{{\mathcal E}}\) is of the Baire class two. If \(f: <0,1>\to R\) is in the Baire class \(\alpha\) and \({\mathcal E}=\{E_ x: x\in <0,1>\}\) is a continuous system of paths, then \(f'_{{\mathcal E}}\) is in the Baire class \(\alpha +2\) if it exists and \(\bar f'_{{\mathcal E}}\) is Lebesgue measurable.