Derivation and Integration on a Fractal Subset of the Real Line (original) (raw)

The ordinary calculus is usually inapplicable to fractal sets, therefore we introduce the various approaches made so far to describe the theory of derivation and integration on a fractal set. In particular we study the Riemann type integrals (s-Riemann integral, s-HK integral, s-first return integral) defined on a closed fractal subset of the real line with finite positive s-dimensional Hausdorff measure (s-set) with particular attention to the Fundamental Theorem of Calculus. Moreover we pay attention to the relation between the s-HK integral, the s-first return integral and the Lebesgue integral respectively. Finally we give a descriptive characterization of the primitives of a s-HK integrable function.