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How to prove a set is unbounded

  • Thread starter happybear
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  • #1
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Homework Statement



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?

Homework Equations



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

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
 
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Answers and Replies

  • #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, ...)
 
  • #3
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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?
 
  • #4
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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?
 
  • #5
HallsofIvy
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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?
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].
 
  • #6
HallsofIvy
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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?
The problem was not to prove that functions in the set are bounded but that the set itself is.
 
  • #7
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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].


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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