Full Idea
Divide points into left and right set. They're 'continuous' if that point is either last member of left set, and greatest lower bound of right (so no least member), or least upper bound of left set (so no last member) and first member of right set.
Gist of Idea
Points are 'continuous' if any 'cut' point participates in both halves of the cut
Source
Harré,R./Madden,E.H. (Causal Powers [1975], 6.IV)
Book Reference
Harré,R/Madden,E.H.: 'Causal Powers: A Theory of Natural Necessity' [Blackwell 1975], p.111
A Reaction
The best attempt I have yet encountered to explain a Dedekind Cut for the layperson. I gather modern mathematicians no longer rely on this way of defining the real numbers.
Related Idea
Idea 10213 Real numbers are thought of as either Cauchy sequences or Dedekind cuts [Shapiro]