A Jordan curve of positive area

William F. Osgood
1903 Transactions of the American Mathematical Society  
The most general continuous plane curve without multiple points may be defined as a set of points which can be referred in a one-to-one manner and continuously to the points of a segment of a right line, inclusive of the extremities of the segment, if the curve is not closed ; and to the points of the circumference of a circle, if the curve is closed.f Such a curve is called a Jordan curve. It may be represented analytically by the equations x=(t), y = yjr(t), where , yjr are single valued
more » ... nuous functions of t in the interval 0 = t = 1 and where the equations »' = <£(*)> y' = f(t), x , y being given, admit at most one solution in the interval 0 = t = 1, when the curve is not closed. In the case of a closed curve
doi:10.1090/s0002-9947-1903-1500628-5 fatcat:bidy5rfusrcxfmbdqds5t7zyue