How to prove a set is unbounded

  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
  2. jcsd
  3. HallsofIvy

    HallsofIvy 41,255
    Staff Emeritus
    Science Advisor

    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. 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. 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. HallsofIvy

    HallsofIvy 41,255
    Staff Emeritus
    Science Advisor

    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. HallsofIvy

    HallsofIvy 41,255
    Staff Emeritus
    Science Advisor

    The problem was not to prove that functions in the set are bounded but that the set itself is.

  8. 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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