Re: Dense vs. Continuous

The poster formerly known as Colleyville Alan a écrit :

"Dave L. Renfro" <renfr1dl@xxxxxxxxx> wrote in message news:8ad65a97-db49-465d-a88b-2be9a244e1e8@xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx
Dave L. Renfro wrote (in part):

I have the book, and I'm somewhat
familiar with it, but it's at home and I'm not.

Oh, I fogot. It's freely available in digital form now.
So let me rephrase this as "I'm not sufficiently motivated
in resolving the issue of whether the authors used "continuous"
in the context of describing sets to pursue the matter".

Dave L. Renfro

To clarify, the authors never mentioned the term "continuous", they mentioned that the rationals were "dense". I was trying to ascertain what the difference was between "dense" and "continuous".

But what on earth make you believe that the term "continous" has any meaning in this context ? "Dense" is a very classical term applied to subsets of a topological space. "Continuous" is only (classiccally) applied to functions. Your questions seem, to be the least, badly formulated. For any comparison, you would at least have to define *your* terms, as you admit yourself that they are not (well, for "continuous") the author's term


I was
(am)uncertain as to whether even trying to compare "dense" and "continuous" is a valid question or if it is an apples-to-oranges type of question.

So, let me try again. If I have a function y = f(x) = x, the graph would be a 45* line through the origin in a standard Cartesian grid. The graph would be a continuous line

You perhaps mean a connected set, or a path-connected set. "Continuous" has no clear meaning in this context.


with no breaks since it would be
using the reals as the domain assuming that x is a random variable

Again, you are using words out of their context. A "random variable" is not what you mean.


and
that the function holds for {x|x E R }

But, suppose I defined the x-axis to be the set of rationals, not the set of reals (assuming it makes sense to do that - please inform me if that is an idiotic concept).

You are now speaking of a Q to Q map


If I were to graph the same function and
said the domain was the set of rationals, would the function be considered a continuous function?

Of course. It is *never* (in math) a matter of opinion, but of definitions and proofs. For any topological set A, the identity function x ->x from A to A is continuous.



On the one hand, in the interval between 1.41421 and 1.41422, the point equal to the square root of 2 would be missing from such a graph, suggesting to me that it is not continuous.

What you mean is that *as a subset of the plane*, the graph is not "continuous" (actually not connected) ; this is correct.


On the other hand, if the
set of rationals is dense and if dense means that you can get infinitely close to the square root of 2 using rationals, then perhaps the graph is considered to be continuous.

No. 1) a set is never said "continuous" (in standard terminology), 2) nothing is ever 'considered' (well, except for borderline-cases like the styatus of the empty set, and even then), but proved or disproved.


I simply have no idea one way or the other
and I have no idea whether the concept is even sensible. But hopefully, it is now clear that I am trying to develop an intuitive understanding of what dense means

Well, you should work in the opposite direction. Learn the definition, see if they correspond to your intuition, and eventually develop ifferent definitions to suit your needs, as the consequences of the definitions are already fixed and, for instance, you have not even yet realized that "dense" has no meaning without an external set of reference (for instance, your graph above is *not* dense in the whole plane).



and how it is or is not related to the concept of
continuous, whether through functions, through sets or through some other thing. If those two concepts belong in two different arenas, that would be worth knowing as well.

Thanks for your patience.
Alan


.

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  • Re: Dense vs. Continuous
    ... mentioned that the rationals were "dense". ... function on the reals is a continuous function. ... If I were to graph the same function and said the domain ...
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  • Re: Dense vs. Continuous
    ... I was trying to ascertain what the difference was between "dense" and "continuous". ... The graph would be a continuous line with no breaks since it would be using the reals as the domain assuming that x is a random variable and that the function holds for ... But, suppose I defined the x-axis to be the set of rationals, not the set of reals. ... On the one hand, in the interval between 1.41421 and 1.41422, the point equal to the square root of 2 would be missing from such a graph, suggesting to me that it is not continuous. ...
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