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Uniform continuity, bounded subsets

  1. Jun 11, 2007 #1
    1. The problem statement, all variables and given/known data

    Show that if f: S -> Rn is uniformly continuous and S is bounded, then f(S) is bounded.

    2. Relevant equations

    Uniformly continuous on S: for every e>0 there exists d>0 s.t. for every x,y in S, |x-y| < d implies |f(x) - f(y)| < e

    bounded: a set S in Rn is bounded if it is contained in some ball about the origin. That is, there is a constant C s.t. |x|<C for every x in S.


    3. The attempt at a solution

    I understand the idea of the proof pretty well but I cannot write the correct mathematical interpretation of it down.

    Basically, S is bounded, so it can be divided into segments (TA called them partitions which is confusing since S is in Rn not R).
    Each segment can be made smaller than d. Then, by uniform continuity we know that f(segment) is smaller than e (bounded) so we can draw a ball around it.
    Since there is a finite number of segments, there are a finite number of balls f(segment). Hence, we can draw a bigger ball around all of them, and thus f(S) is bounded.

    How do I put this into math symbols?

    Thanks for your help =)
     
  2. jcsd
  3. Jun 12, 2007 #2

    NateTG

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    Homework Helper

    The thing you want here is a cover, and not a partition.

    Can you show that:
    [tex]N_{\epsilon}(f(x)) \supset f(N_{\delta}(x))[/tex]
    and that
    [tex]S \subset \bigcup_{\rm{finite}} N_{\delta}(x_i)[/tex]

    As a note: When you're applying a function to each of a set of objects, it's typical to refer to the result as the image of applying the function to that set. Something like:
    [tex]N_{\epsilon}(f(x)) \supset Im_f(N_{\delta}(x))[/tex]
    might be better notation, because errors can occur when sets are treated like single-valued objects.
     
    Last edited: Jun 12, 2007
  4. Jun 14, 2007 #3
    What's N_e?
     
  5. Jun 14, 2007 #4

    NateTG

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    The epsilon neighborhood of something. It's the same idea as, say:
    [tex]B(\epsilon,x)[/tex]
    The ball of radius [itex]\epsilon[/itex] around [itex]x[/itex].

    I probably didn't make the best choices in notation there -- my apologies.
     
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