Dense vs. Continuous

I am reading through Elementary Real Analysis by Thompson, Bruckner & Bruckner. It is well-written, user friendly, and free!

They discuss the fact that the rational numbers are dense and describe the concept thusly:

"The rational numbers are dense. They make an appearance in every interval; there are no gaps, no intervals that miss having rational numbers."

And later, they state

"...every real is as close as we please to a rational..."

Finally, they mention:

"For theoretical reasons this fact is of great importance too. It allows many arguments to replace a consideration of the set of real numbers with the smaller set of rationals."


Now intuitively, I understand that the real numbers includes the rational and the irrational and so it seems to make sense to say in the third quoted sentence "It allows many arguments to replace a consideration of the set of real numbers with the smaller set of rationals" (the set of real numbers is larger than the rationals).

Ok, the rationals are countably infinite but dense and the real numbers are uncountably infinite. But if, looking only at the rational numbers, we can always find an infinity of rational numbers between any two given rational numbers, how can we say that the set is not continuous?

Yes, it is "missing" the irrationals and is a smaller set than the real numbers, but from the definition of "dense", there does not seem to be any "holes" in the set. I think that I understand the meaning of "dense", from the author's description, but perhaps I am not correctly understanding what the term implies. Any thoughts?

Thanks.
Alan



p.s. if anybody is looking for a free Analysis textbook, they authors have both an elementary and a graduate level text available for free download here:
http://classicalrealanalysis.com/download.aspx


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Relevant Pages

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  • Re: counter example in analysis
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