Some fundamental results of calculus on fractal sets (Q1288339)

In this paper the authors introduce the definitions of \(s\)-derivative and \(s\)-integral on fractal sets contained in the set of real numbers. The \(s\)-derivative and \(s\)-integral are defined in an analogous fashion as the classical derivatives and integrals, but with the Hausdorff measure taking over the role of the distance. Several simple facts and examples are given, including (1) any bounded variation function on a fractal set \(E\) is \(s\)-differentiable provided the Hausdorff measure \({\mathcal H}^s(E)\) of \(E\) is finite and \(0<s\leq 1\); (2) the \(s\)-derivative of the Cantor function is infinite at every \(x\in C\), where \(C\) is the classical two third Cantor set.