A solution for the differences in the continuity of continuum among mathematicians

Abstract: There are the longstanding differences in the continuity of continuum among mathematicians. Starting from studies on a mathematical model of contact, we construct a set that is in contact everywhere by using the original idea of Dedekind's cut and weakening Order axioms to violate Order axiom 1. It is proved that the existence of the set constructed can eliminate the differences in the continuity.