Fun little thing I'm curious about.
From reading books and stuff, everyone says that "ccompletenesss is the important thing about the real numbers that gives them the properties necessary fro calculus.
My fun little question is this: do the real numbers need completeness to have the intermediate value theorem?
To put my question more concisely, suppose we have a function which maps the rational numbers into the rational numbers. Also, suppose it is continuous. That is, for any positive RATIONAL epsilon, there is a positive RATIONAL delta such that... you get the picture. My question is this: does the intermediate value theorem hold? Counter-example, anyone? I've been trying to come up with one.
The point I'm trying to make here is that the epsilon-delta definition of continuity seems kind of...independent of the completeness axiom. So, yea. Would the intermediate value theorem hold for a rational-to-rational function, with continuity defined as above?