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Show the following is a metric

  1. Dec 8, 2013 #1
    1. The problem statement, all variables and given/known data

    Show that ##d(f,g) = \int_{0}^{1}\left | f(x) - g(x) \right | dx## is a distance function. Where ##f : [0,1] \rightarrow R## and ##f## is continuous.

    2. Relevant equations



    3. The attempt at a solution
    I am stuck on the second property where you have to show d(f,g) = 0 iff f = g. THe left direction is trivial. However d(f,g) = 0 implying f=g is giving me trouble. I have tried contrapositive, but it doesnt seem to be getting me anywhere.
     
  2. jcsd
  3. Dec 8, 2013 #2

    Dick

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    The fact you need is that if F(x)>=0 on [0,1], F(a)>0 for some ##a## in [0,1] and F(x) is continuous (very important) then ##\int_{0}^{1}\left | F(x) \right | dx \gt 0##. Can you figure out how to prove that?
     
    Last edited: Dec 8, 2013
  4. Dec 8, 2013 #3
    wouldn't I need to show that ##\int_{0}^{1}\left | F(x) \right | dx > 0##
    to get the contrapositive?
     
  5. Dec 8, 2013 #4
    Oh I see! ok. NVM. Gonna try to prove it
     
  6. Dec 8, 2013 #5

    Dick

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    Yes, of course. Typo. Sorry. I corrected it.
     
    Last edited: Dec 8, 2013
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