Peter,

It seems to me that Weyl offered a variety of approaches, and I wonder if you wouldn't mind being more specific either as to source or about which of his ideas you have in mind.

For example in Space, Time, Matter, he employs the idea of a tensor calculus. I'd like to take a moment to look at tensors and ask myself if they are a continuum. 

First of all, a tensor is a mathematical entity, and it strikes me that anything called an "entity" is closed rather than continuous. Jessica Wilson and Jeremy Butterfield have done some work on this. An entity is defined by its intrinsic properties, whether they be essential or accidental. This means the unobservables beyond that enclosure are not relevant, so an entity by definition cannot be not part of a continuum.

A tensor or vector presumes a point in Cartesian space-time that anchors a magnitude pointing elsewhere in Cartesian space. To "point elsewhere" is not a continuum, but a specific value. Points on a pseudo-continuum, as in calculus, remain points, albeit infinitesimally small. They only approximate a continua, just as movie stills approximate motion. More broadly any presumption of a Cartesian space {an Aristotelian "substance" having a location defined by the three conventional dimensions and possessing a set of properties) seems to presume an ontological closure right from the start, for the entity is defined by what is local, whatever the origin of its properties may be. 

This does not apply to extrinsic properties such as ontic probability values. Without arguing the case, I believe a representation of a continuum can only be done through extrinsic properties.

Even more broadly and boldly, it seems the modern Western world view is incompatible with continua. One might look at Buddhism or Daoism for an alternative. Rather than start by presuming a sacrosanct self at a spatio-temporal point and lending the world meaning relation to it (as advocaed by Leon Battista Alberti) the world is a flux, and he self is secondary, an entity that imposes limits on that flux (kind of an explicit symmetry break)..

A tensor is a mathematical representation of the relation of mathematical entities, whether they be scalars, vectors or other tensors. Is this a structuralism that would have relations exist independently of their nodes? If so, it seems to imply an objective idealism or reification that I suspect is contrary to the intent of the OP. That is, if we reify relations, then of course we have continua, but the price we pay is superstition. In other words, reified abstractions by definition transcend particulars.

I suspect a continuum is beyond the reach of folk psychology/science. It seems the consensus in neuroscience that the signal from sensory organs is analog in origin but converted to a digital signal so that it can be cognitively processed. Mathematics goes beyond folk psychology, but I doubt that a mathematical representation of a continuum is itself a continuum. I also suspect that any representation of a continuum must, perhaps to borrow from Leibiz, continuum must essentially joint both the whole and its parts. .