If you are asking me that question, I assumed a continuous universe is one in which a line could exist. So since modern science believes an infinite line (a rather redundant term, I realize) cannot actually exist, then the house of cards upon which the ancient Greeks built their IP has been discounted long ago. My grandfather actually took the subject "Euclid" in elementary school, but I am sure no one has taken that for 100 years.
Many scientists hypothesize that the universe is infinite, so an infinite line can in fact exist in it.
Plus Euclidean geometry is like the basic geometry from which most education and theories sprout.
Sorry, I do know what my grandfather took in school. My point was that if FE'ers cling to a belief that our current system including a round earth is simply a mistake based on a house of cards that could collapse -- they should know that the ancient Greek house of cards collapsed a long time ago. People no longer take Euclid in school because there is more to learn now than what the ancient Greeks had figured out. The house of cards that Newtonian physics was based on has been smashed in the sense that it is insufficient to explain events outside our daily experience such as how galaxies move or how subatomic particles behave. But no engineer uses relativistic formulas to design a bridge, because Newtonian principles are good enough to assure that people will not die from a collapse of the bridge. I have not heard any scientist in the last 50 years say that they believe the universe is infinite at the present, or rather that we can see the radiation left from when the big bang got big enough for light to travel and this barrier forms the limit of our universe. I am trying (I will use different words to see if I can be clearer) to say that the chance that we will discover a better system of viewing the world that ends up showing the earth is actually flat ain't gonna happen. String theory, multi-universe theory, whatever, is not going to suddenly reveal that we are wrong about the earth being a sphere and that the sun is 3000 miles up and that it bounces around and moves between the tropics and the earth is accelerating up and that the air is being held above this disk by some magical barrier or wall or something etc etc.