Computers are just devices that perform computations. It's perfectly possible to construct computers that operate on values that are continuous and not discrete. In fact, such computers have been constructed before. They are know as analog computers and were used for things like numerically solving differential equations.
See here for a concrete application of the theory: https://icfp19.sigplan.org/details/icfp-2019-papers/15/Sound...
Another application is the Android calculator, which computes over exact real numbers in the sense described by Computable Analysis.
Finally, it's debatable whether analog computation achieves true computation over the real numbers. The presence of physical noise, and the fact that many physical quantities (like electrical charge) are ultimately discrete at the quantum level, implies that it doesn't really work AFAICT.