The concept of continuity that Peirce highlights in his synechism is a logical principle that is somewhat more general than the concepts of either mathematical or physical continua.
Peirce’s concept of continuity is better understood as a concept of lawful regularity or parametric variation. As such, it is basic to the coherence and utility of science, whether classical, relativistic, quantum mechanical, or any conceivable future science that deserves the name. (As Aristotle already knew.)
Perhaps the most pervasive examples of this brand of continuity in physics are the “correspondence principles” that describe the connections between classical and contemporary paradigms.
The importance of lawful regularities and parametric variations is not diminished one bit in passing from continuous mathematics to discrete mathematics, nor from theory to application.
Here are some further points of information, the missing of which seems to lie at the root of many recent disputes on the Peirce List:
It is necessary to distinguish the mathematical concepts of continuity and infinity from the question of their physical realization. The mathematical concepts retain their practical utility for modeling empirical phenomena quite independently of the (meta-)physical question of whether these continua and cardinalities are literally realized in the physical universe. This is equally true of any other domain or level of phenomena — chemical, biological, mental, social, or whatever.
As far as the mathematical concept goes, continuity is relative to topology. That is, what counts as a continuous function or transformation between spaces is relative to the topology under which those spaces are considered and the same spaces may be considered under many different topologies. What topology makes the most sense in a given application is another one of those abductive matters.