One-to-one continuous images of the reals play an important role in dynamical systems as all non-periodic orbits fall in this class. We present a characterization of one-to-one continuous images of the reals in the realm of hereditarily unicoherent spaces. The characterization is reminiscent of the well-known characterization of the reals which requires each point to be a cut point of order 2. As a particularly useful corollary we obtain a characterization of the non-compact one-to-one continuous images of the halfline. © 1991.