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Why d is not a metric

  • Thread starter happybear
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  • #1
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Homework Statement



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

Homework Equations





The Attempt at a Solution

 

Answers and Replies

  • #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.
 
  • #3
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why infinity is not a map into R?
 
  • #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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