In an earlier post I state that a fluid is not a continuum, but a fractal disposition of contiguous regions at all scales. Elsewhere I state that a region is continuous until it is not! The reason for these statements is to highlight the abstract nature of the notion of continuous, and the linking to differentiability.
2 lines may be discontinuous and discrete, or continuous and discrete. We call the second case contiguous. We may make these abstract forms collinear etc but that does not make them identical unless we define that case. In fact I have had to define collinearity as identity in order to justify that a constructed diameter to a circle, that is rigorously a constructed set of dual points, passing through the centre of a circle actually goes through a unique dual point on either side of the circle. Unless collinearity is defined as identity I can only point to the case by case verification by drawing.
Thus a contiguous continuity may be collinear with a continuous line . If I make both lines identical I am blurring the distinction between contiguous and continuos. I prefer to call it what it is, an imposed description on the same line.
Contiguity for a surface or a solid is easier to distinguish. A line crossing an imposed boundary reveals whether the surface is continuous or contiguous by where it continues after the boundary! Surfaces and solids can be contiguous along a line or plane, but shifted along the boundary of contact. In a solid this is associated with a break in the medium, allowing some other substance to intersperse, but in a fluid, such a break is not necessary. A fluid is to be characterised by these instantaneous boundary shifts where contiguity is preserved but continuity is not.