Area-filling curves

Abstract In this paper, we study area-filling curves, i.e. continuous and injective mappings defined on [0, 1] whose graph has positive measure. Current literature calls them “Osgood curves”, but their invention is due to H. Lebesgue. Stromberg Tseng constructed homogeneous curves offered an elegant example. We show that appropriate variant of Knopp’s construction attains the same homogeneity result. Section 4, discuss briefly existence “invasive” curve, a mapping from half-open interval 1[ unit square, image measure 1. last section, several aspects Lance-Thomas connecting it with other Tseng.