[mbeex]

Show me a mathematical continuum in the real-world. And this is just the domain, nothing said about the mapping yet (the continuous function). Every analogy has its limits.

[tomrod]

The space represented in a flat piece of paper.

The space is continuous even if we can only measure down to the Planck length.

[mbeex]

Key here: "represented"

I'm only a mathematician, no physicist. But I think to remember, that the concept of a continuous physical space becomes quite muddled at this scale.

[nerdponx]

"The space represented by the water in a convex container"

[hgsgm]

If you get to assume continuous paper, then I get assumed continuous paper.

If you don't want to assume continuity, then you get it back by rephrasing your theorems as "within a margin of error equal to the distance between discrete objects".