On the Peano curve of Lebesgue

Abstract

Let (/) be the even continuous function of period two which is defined in the interval (0, 1) as follows : (t) = 0 in (0, 1/3), (/) = 1 in (2/3, 1), and (/) is linear in (1/3, 2/3). Our curve is defined by the parametric equations *(0 = y 7(0 + ^/(3 2 /) + j t /(30 + , (1) ill yW = y/(3*) + -/(3 3 0 + -/(3o + , o^/gi.