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Proving something is closed

  1. Jul 14, 2011 #1
    1. The problem statement, all variables and given/known data
    prove that the set of continous functions on [0,1] that are increasing is a closed set.


    2. Relevant equations



    3. The attempt at a solution
    Need to prove the complement is open. So need to prove the set of continous functions on [0,1] that are non increasing is open. Let f be a continous function that is non increasing on [0,1]. need to choose an epsilon st B(f)={g in C([0,1]): sup|g(x)-f(x)|<epsilon} is contained in the set of continous functions on [0,1] that are non increasing.
     
  2. jcsd
  3. Jul 14, 2011 #2

    micromass

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    Hi l888l888l888! :smile:

    So f is nonincreasing. So let [a,b] be an interval such that f(b)<f(a). How close must a function g be such that also g(b)<g(a)??

    As a further hint, you must find an [itex]\varepsilon >0[/itex] such that every element in [itex][f(b)-\varepsilon,f(b)+\varepsilon][/itex] is smaller than [itex]f(a)-\varepsilon[/itex], do you see why?
     
  4. Jul 14, 2011 #3
    well i narrowed it down to a few cases. assuming we have an interval [f(b),f(a)] ...
    if g(b)<f(b) then |g(a)-f(b)| has to be less than |g(b)-f(b)|. if f(b)<g(b)<f(a) then |g(a)-f(a)| has to be less than |g(b)-f(a)| and the last case for g(b)> f(a) i have not figured out. im basically going about this by looking at this graphically and trying to write it out as a proof
     
  5. Jul 14, 2011 #4

    micromass

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    You have to eliminate the last case by choosing [itex]\varepsilon[/itex] small enough...
     
  6. Jul 15, 2011 #5
    well in answer to the hint. in order for everything in [f(b)-epsilon, f(b)+epsilon] to be less than f(a)-epsilon, f(b)+epsilon has to be less than f(a)-epsilon. rearranging this inequality we get f(a)-f(b)>2epsilon. ==> (f(a)-f(b))/2 > epsilon ==> epsilon + some number c = (f(a)-f(b))/2 ==> epsilon= (f(a)-f(b))/2 - c.
     
  7. Jul 15, 2011 #6

    micromass

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    Indeed, this epsilon will make it all work! Btw, it's perfectly ok to say to choose [itex]\varepsilon<\frac{f(a)-f(b)}{2}[/itex]. You don't need to introduce the number c, but it's not wrong if you do.
     
  8. Jul 15, 2011 #7
    oh ok thanks so much. Your so helpful!!!!!!!!!!!! :)
     
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