The study of temporal networks in discrete time has yielded numerous insights into time-dependent networked systems in a wide variety of applications. For many complex systems, however, it is useful to develop continuous-time models of networks and to compare them to associated discrete models. In this paper, we study several continuous-time network models and examine discrete approximations of them both numerically and analytically. To consider continuous-time networks, we associate each edge in a graph with a time-dependent tie strength that can take continuous non-negative values and decays in time after the most recent interaction. We investigate how the mean tie strength evolves with time in several models, and we explore -- both numerically and analytically -- criteria for the emergence of a giant connected component in some of these models. We also briefly examine the effects of interaction patterns of our continuous-time networks on contagion dynamics in a susceptible-infected-recovered model of an infectious disease.