Continuity is among the oldest of mathematical problems.
From a technical standpoint most issues were resolved long ago by defining continuous structures as limiting cases of discrete structures as the elements in the discrete structure increase without limit. Those question how it is possible to construct a continuous interval from an infinite number of infinitesimal intervals. There is no mathematics to do that. It is a discrete algebra because it operates on discrete abstractions like lines, planes, points and circles and not on a completed infinite collection of infinitesimal objects. All we do is show that certain properties can be made to hold with arbitrarily high accuracy if we can construct a sufficiently large discrete system.