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How to prove a set is unbounded 
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#1
Apr2909, 03:56 PM

P: 19

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 


#2
Apr2909, 04:01 PM

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PF Gold
P: 39,495

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 f_{n}(x)= n. for n= 1, 2, 3, ...) 


#3
Apr2909, 04:36 PM

P: 19

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
Apr2909, 04:58 PM

P: 238

How to prove a set is unbounded
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
Apr2909, 05:59 PM

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P: 39,495

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
Apr2909, 06:00 PM

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PF Gold
P: 39,495




#7
Apr2909, 08:03 PM

P: 19

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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