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Why d is not a metric

  1. May 11, 2009 #1
    1. The problem statement, all variables and given/known data

    If d= sup{|f(x)-g(x)|} where f, g are contuous function from R to R, why is d not a metric

    2. Relevant equations



    3. The attempt at a solution
     
  2. jcsd
  3. May 11, 2009 #2

    quasar987

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    Take f(x)=x, g(x)=0. then d(f,g)=infinity. So d is not a map into R.
     
  4. May 11, 2009 #3
    why infinity is not a map into R?
     
  5. May 11, 2009 #4

    quasar987

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    I said that d is not a map into R. And this is because if you take f(x)=x and g(x)=0, then d(f,g)=sup{|x|:x in R} is not an element of R. Indeed, the real number sup{|x|:x in R}, if it exists, is an upper bound for the set {|x|:x in R}. But this set is unbounded from above and hence possesses no upper bound so sup{|f(x)|:x in R} does not exists.
     
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