In the third, fourth and fifth centuries there flourished on an amazing scale the practice of revering the relics and shrines of the martyrs. The care of the dead was a visible and shocking demonstration to the pagan world of Christian belief. Traditionally, historians have accounted for this as an adoption by Christians of pagan practices. Addressing this phenomenon, Peter Brown writes in his The Cult of the Saints - Its Rise and Function in Latin Christianity -
...we must remember that the Christian church had risen to prominence largely because its central ritual practices and its increasingly centralised organisation and financial administration presented the pagan world with an ideal community that had claimed to modify, to redirect, and even to delimit the bonds of the kin. The church was an artificial kin group. Its members were expected to project onto the new community a fair measure of the sense of soldarity, of the loyalties, and of the obligations that had previously been directed to the physical family. Nowhere was this made more plain than in the care of the dead. p. 31
Christians began to have their own cemetries, and to note carefully the anniversaries of the deaths of martyrs and bishops. By the time of the Constantinian turn, with the growth in Christian belief amongst the wealthy elites, this meant for the appearance en masse of endowed shrines and cemetries all over the empire. As Brown writes:
What we have seen is not the growth of new beliefs within the Christian communities, but the restructuring of old beliefs in such a way as to allow them to carry a far heavier charge of public meaning... p. 48
The continuity was not with pagan popular religion, but in fact with Christian belief of the previous eras. This was not merely a feature of popular religion either, but was encouraged and sponsored by the lay and clerical elites. Noticeably, the reverence for the martyrs outlasted the persecutions which made them: the churches were insistent that this was a self-identifying practice of utter necessity for who they were as a group.