  In this paper we investigate the space of $\mathbb{R}$-places of an algebraic
function field of one variable. We deal with the problem of determining when
two orderings of such a field correspond to a single $\mathbb{R}$-place. To
this end we introduce and study the space of cuts on a real curve and prove
that the space is homeomorphic to the space of orderings. Finally, we prove
that two cuts (consequently, two orderings) correspond to a single
$\mathbb{R}$-place if they are induced by a single ultrametric ball.
