  Many natural real-valued functions of closed curves are known to extend
continuously to the larger space of geodesic currents. For instance, the
extension of length with respect to a fixed hyperbolic metric was a motivating
example for the development of geodesic currents. We give a simple criterion on
a curve function that guarantees a continuous extension to geodesic currents.
The main condition of our criterion is the smoothing property, which has played
a role in the study of systoles of translation lengths for Anosov
representations. It is easy to see that our criterion is satisfied for almost
all the known examples of continuous functions on geodesic currents, such as
non-positively curved lengths or stable lengths for surface groups, while also
applying to new examples like extremal length. We use this extension to obtain
a new curve counting result for extremal length.
