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
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, ...)
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 [tex] f : [a, b] \to R [/tex] and let A = [a, b] is bounded, is it necessarily true that f(A) is bounded?
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].
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