This is my own question but I assume it has been asked before; I just have not found where. I am curious of the answer but I don't know how to approach solving it. I would appreciate it if someone could help me out with tags because I don't have any experience with this area in math. Also, please let me know if I misued the words "connected space." I was originally going to use "continuous" but saw this question and changed it.
This is my original question
Can any continuous function be represented by infinite polynomial interpolation of random points in a definite interval?
Which I simplified to
Is every point on a line of definite length covered after infinitely many points are placed on it?
Which I realized is equivalent to
Is a continuous line formed by an infinite amount of points on the x-axis contained in a definite interval?
Which I simplified down to the question of this post.
Depending on the answers to this question, I plan to ask my original question along with some follow-ups about extending it to an infinite interval and to parametric functions.