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Math Help - Partial Derivatives and Continuous Functions

  1. #1
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    Partial Derivatives and Continuous Functions

    Q. For what values of p is the function

    f(x,y) = { (x+y)^p/(x^2 +y^2) where (x,y) can not equal (0,0)
    0 else


    continuous on R^2

    I'm finding this problem very confusing. Please help. I know that 0 <= (x-y)^2 =
    x^2 -2*x*y+y^2 so 2*x*y <= x^2 +y^2

    and I have to use the inequality
    (x+y)^2 = x^2 +2*x*y +y^2 <= 2*x^2 +2*y^2

    How do I proceed from here. Thanks in advance.
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  2. #2
    A Plied Mathematician
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    Well, you could always use the calculus definition of continuity: the limit of a function must equal its value in any interval in which the function is continuous. You're going to have two issues that I can see: negative values of x+y combined with a fractional exponent p such as 1/2, and making the sure the limit at the origin matches the function value at the origin. These two issues will, I think, restrict the possible values of p. See what that gives you.
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