Counter-example: Let over the interval from Now, and By the Intermediate Value Theorem, you'd expect there to be a rational root in the interval. But there isn't.

In addition, the delta-epsilon definition of continuity is entirely equivalent to the "calculus" definition, where the limit of a function at a point is equal to the function's value at that point. All you have to do is look at an irrational function value, such as to see that if the irrationals are not included, you can't have continuity.