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Set theory and analysis: Cardinality of continuous functions from R to R

  1. Sep 22, 2009 #1
    1. The problem statement, all variables and given/known data
    Prove the set of continuous functions from R to R has the same cardinality as R


    2. Relevant equations
    We haven't done anything with cardinal numbers (and we won't), so my only tools are the definition of cardinality and the Schroeder-Bernstein theorem and its consequences.

    I also don't know any "high brow" mathematical facts about continuous functions.
    3. The attempt at a solution
    Not much. By Schroeder-Bernstein, we need an injection [tex]f:\mathbb{R}\rightarrow C^0[/tex] and vice versa. We have the usual embedding map from [tex]\mathbb{R}[/tex] to [tex]C^0[/tex], but I've no idea how to construct an injection going the other way. Given any continuous function, how do I uniquely identify it with a real number?
     
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  3. Sep 22, 2009 #2

    Dick

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    I think you need to know at least a few things about cardinality. The main trick is that every continuous function is determined by its values on the rationals, which is a countable set. Does that help?
     
  4. Sep 22, 2009 #3
    ah yes, then I should be able to associate any continuous function with a real number in [0,1) whose digits correspond to the value of the function evaluated at every q in Q. doing that shouldn't be too hard.

    thanks!
     
  5. Sep 22, 2009 #4
    as an aside: how do I know that every real number is the limit of some sequence of rational numbers? I mean, I "know" that this is pretty much what R is (as the completion of Q), but I'm not sure how to rigorously back that up.
     
  6. Sep 22, 2009 #5

    Hurkyl

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    Isn't "every real number is a limit of rational numbers" pretty much (among other things) literally what "R is the completion of Q" means?
     
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