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Math Help - Metric Space

  1. #1
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    Metric Space

    I'm trying to show that, given X is the set of all continuous functions  f:[a,b] \rightarrow R , and d(f,g) =  \int_{a}^{b} | f(t) - g(t) | dt , that (X, d) defines a metric space.

    My confusion is coming in when I let a = 0, b =  \frac{\pi}{2} , and f = sine and g = cosine. Then wouldn't the distance function between these two be

     \int_{0}^{ \frac{\pi}{2}} | sin(t) - cos(t) | dt
     = - (cos(\frac{\pi}{2}) - cos(0)) - ( sin(\frac{\pi}{2}) - sin(0))
     = -(0 - 1) - (1 - 0) = 1 - 1 = 0

    But f != g, so isn't that a problem?
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  2. #2
    MHF Contributor Bruno J.'s Avatar
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    What did you make of the absolute value?

    Your calculation is wrong. The integral of a non-negative continuous function is 0 if and only if the function is identically 0. The function you are integrating is clearly non-negative and continuous, but it's not identically 0!
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