I read somewhere that in minimal geometry(incidence, betweenness and congruence axioms) the circular continuity
If a circle has one point inside and one point outside another circle, then the two circles intersect in two points.
implies the elementary continuity
If one endpoint of a segment is inside a circle and the other endpoint is outside then the segment intersects the circle.
but that the converse is not true.
Could you provide a (preferably simple) model to support the last claim?