We present a continuous threshold model (CTM) of cascade dynamics for a network of agents with real-valued activity levels that change continuously in time. The model generalizes the linear threshold model (LTM) from the literature, where an agent becomes active (adopts an innovation) if the fraction of its neighbors that are active is above a threshold. With the CTM we study the influence on cascades of heterogeneity in thresholds for a network comprised of a chain of three clusters of agents, each distinguished by a different threshold. The system is most sensitive to change as the dynamics pass through a bifurcation point: if the bifurcation is supercritical the response will be contained, while if the bifurcation is subcritical the response will be a cascade. We show that there is a subcritical bifurcation, thus a cascade, in response to an innovation if there is a large enough disparity between the thresholds of sufficiently large clusters on either end of the chain; otherwise the response will be contained.