Re: Dense vs. Continuous


"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". 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 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 {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). 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?

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. 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. 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 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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Relevant Pages

  • 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 ...
    (sci.math)
  • Re: Dense vs. Continuous
    ... I was trying to ascertain what the difference was between "dense" and "continuous". ... The graph would be a continuous line ... has no clear meaning in this context. ... But, suppose I defined the x-axis to be the set of rationals, not the set of reals. ...
    (sci.math)
  • Re: Dense vs. Continuous
    ... I mean that to draw the graph of this function, you never need to lift your pencil from the paper. ... But, suppose I defined the x-axis to be the set of rationals, not the set of reals. ... equal to the square root of 2 would be missing from such a graph, suggesting to me that it is not continuous. ... So, when I askd if the graph is continuous, I mean are there any gaps in it or is the line unbroken? ...
    (sci.math)
  • Re: Exotic functions (elementary)
    ... A FUNCTION WHOSE GRAPH IS DENSE IN THE PLANE ... nonlinear additive functions can map onto ...
    (sci.math)
  • Re: counter example in analysis
    ... Since I referred to rationals, ... different not just from the irrationals but from the real ones, ... On the other hand, uncountable implies dense. ... If I consider the reals one by one, then they are not dense but countable. ...
    (sci.math)