How to prove a set is unbounded

  1. Apr 29, 2009 #1
    1. The problem statement, all variables and given/known data

    Suppose I have a set E that contains all continuous function f from [a,b] --> R, I think this is unbounded, but can I prove it?

    2. Relevant equations

    d(f1,f2)= sup{f1(x)-f2(x)}

    3. The attempt at a solution
    I want to show |d(f1,f2)|>M for some M, but I don't know if this is the right direction and how to do it
     
    Last edited: Apr 29, 2009
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  3. Apr 29, 2009 #2

    HallsofIvy

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    A set is "unbounded" if, for any positive number A, there exist point p and q in the set such that d(p,q)> A. That, of course, depends on how you measure "distance"- d(p,q).
    What is your definition of d(f, g) for f and g in the set of continuous functions from [a, b] to R?

    (What ever your d(f,g) is, consider the functions fn(x)= n. for n= 1, 2, 3, ...)
     
  4. Apr 29, 2009 #3
    But there are no definition of f, that is the point. F will be any continuous function from [a,b] to R. Can I prove this set is infinite?
     
  5. Apr 29, 2009 #4
    If you think that it is false, you can provide a counterexample.

    So, if [tex] f : [a, b] \to R [/tex] and let A = [a, b] is bounded, is it necessarily true that f(A) is bounded?
     
  6. Apr 29, 2009 #5

    HallsofIvy

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    Do you mean d, the distance function? There has to be a definition of distance in order to talk about "bounded" or "unbounded". And you don't want to prove the set is infinite (that's trivial), you want to prove it is unbounded, a completely different thing.

    Is it possible that you are talking about the set of continuous functions from [a,b] to R? For that, the distance function is most commonly [itex]d(f, g)= \max_{a\le x\le b} |f(x)- g(x)|[/itex].
     
  7. Apr 29, 2009 #6

    HallsofIvy

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    The problem was not to prove that functions in the set are bounded but that the set itself is.
     
  8. Apr 29, 2009 #7


    Yes. That is what I meant. And what I want to show is that there are infinitely many functions of f. That is how I interpret the question, since they ask if the set is unbounded. Or do I misunderstand it
     
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