for all such points z, ignoring multiples of 2π. Since the tangent at every point of the half line (x0,∞) has slope 0, it follows that the tangent at every point of the image curve f((x0,∞)) has slope argf0(z) = 0. Hence f((x0,∞)) is a half line parallel to the real axis and has left hand end point f(x0).