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Homogeneity holds but additivity does not. I'm stuck!

  1. Feb 7, 2008 #1
    1. The problem statement, all variables and given/known data
    Give an example of a function f:R^2 -> R such that f(av) = a(f(v)) for all a in R and all v in R^2 but f is not linear.


    2. Relevant equations
    f(v + w) = f(v) + f(w) (Additivity)


    3. The attempt at a solution

    I really can't think of a function that will satisfy these properties. I know that for this to work, homogeneity must hold but additivity must not. I tried several functions, but none worked. Any hints? Thanks.
     
  2. jcsd
  3. Feb 7, 2008 #2

    Dick

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    How about f((x,y))=x^(1/3)*y^(2/3)?
     
  4. Feb 7, 2008 #3
    Thanks! I see how you came up with that now. I tried f((x, y)) = x^2 + y^2 but the first property didn't hold since a changed. :rolleyes: Your function fixes that issue. Thanks again for the help.
     
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