  The possibility that a discrete process can be fruitfully approximated by a
continuous one, with the latter involving a differential system, is
fascinating. Important theoretical insights, as well as significant
computational efficiency gains may lie in store. A great success story in this
regard are the Navier-Stokes equations, which model many phenomena in fluid
flow rather well. Recent years saw many attempts to formulate more such
continuous limits, and thus harvest theoretical and practical advantages, in
diverse areas including mathematical biology, image processing, game theory,
computational optimization, and machine learning.
  Caution must be applied as well, however. In fact, it is often the case that
the given discrete process is richer in possibilities than its continuous
differential system limit, and that a further study of the discrete process is
practically rewarding. Furthermore, there are situations where the continuous
limit process may provide important qualitative, but not quantitative,
information about the actual discrete process. This paper considers several
case studies of such continuous limits and demonstrates success as well as
cause for caution. Consequences are discussed.
