On the completion of Skorokhod space

We consider the classical Skorokhod space $D[0,1]$ and the space of continuous functions $C[0,1]$ equipped with the standard Skorokhod distance $\rho$. It is well known that neither $(D[0,1],\rho)$ nor $(C[0,1],\rho)$ is complete. We provide an explicit description of the corresponding completions. The elements of these completions can be regarded as usual functions on $[0,1]$ except for a countable number of instants where their values vary "instantly".