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Can you find f,g st sup{|f-g|}=0 but f is not equal to g

  1. May 9, 2009 #1
    1. The problem statement, all variables and given/known data
    For all real number x, can you find a function f and g such that
    sup|f(x)-g(x)|=0

    2. Relevant equations



    3. The attempt at a solution
     
  2. jcsd
  3. May 9, 2009 #2

    HallsofIvy

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    What have you done on this problem yourself? In particular, what does sup |f(x)- g(x)| mean? And if you are going to put "f is not equal to g" please put it in the statement of the problem!
     
  4. May 9, 2009 #3

    sup{|f-g|} means that the maximum of the value. I try to find a function f approaching g, but this eem not to be the case
     
  5. May 9, 2009 #4

    HallsofIvy

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    Strictly speaking, sup does NOT mean 'maximum', it means "least upper bound" which may or may not be a maximum. What do you mean "f approaching g"? As x approaches what value? This has to be true over all x.

    Suppose f were NOT equal to g. Then there must exist some x such that [itex]f(x)\ne g(x)[/itex]. Let M= |f(x)- g(x)| for that x. Now, what can you say about sup|f(x)- g(x)|?
     
  6. May 9, 2009 #5
    so does that mean that no matter whether X is compact or not, there is no such a function?
     
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