# How to prove a set is unbounded

• happybear
In summary, the conversation discusses the concept of an unbounded set and how it applies to a set of continuous functions from [a, b] to R. The distance function is defined as d(f, g) = max_{a≤x≤b} |f(x)-g(x)| and the goal is to prove the unboundedness of the set, rather than the boundedness of individual functions within the set. The conversation also clarifies that the set in question is the set of continuous functions from [a, b] to R.

## 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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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, ...)

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?

If you think that it is false, you can provide a counterexample.

So, if $$f : [a, b] \to R$$ and let A = [a, b] is bounded, is it necessarily true that f(A) is bounded?

happybear said:
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 $d(f, g)= \max_{a\le x\le b} |f(x)- g(x)|$.

konthelion said:
If you think that it is false, you can provide a counterexample.

So, if $$f : [a, b] \to R$$ 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.

HallsofIvy said:
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 $d(f, g)= \max_{a\le x\le b} |f(x)- g(x)|$.

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