Can you draw a curve such that it intersects every x at y=0 without ever crossing over itself (that is to say, without hitting the same value twice)?
Wouldn't accomplishing this feat mean infinitely approaching every point on the numberline? Can that be done without a straight horizontal line at y=0?
If so what would stop you from creating a function that does the same, not to just a numberline but a 2 dimensional plane?
If such a line was attempted, isn't there an infinite about of space between all points that a line should never need to cross itself to get out of a spiral it's created?
In order to do this would you need an infinitely complex function? If so is it possible to prove that the function to do this could in principle exist, but we able to prove that since it must be infinitely complex the function can't be known?
In short, is it possible to intersect every infinite point on a plane once and only once with a continuous line?
Am I missing something fundamental about lines and infinities?
Has this kind of question been explored before?