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Multivariable Continuity

  1. Mar 17, 2010 #1
    1. The problem statement, all variables and given/known data
    Define f(0,0)=0 and f(x,y) = x2 +y2-2x2y-4x6y2/(x4+y2)2.

    Show for all (x,y) that 4x4y2<=(x4+y2)2 and conclude that f is continuous.

    2. Relevant equations

    3. The attempt at a solution
    Showing the inequality is trivial, but I do not see how I can conclude the function is continuous. I've done some messing around with the form of f, but am not getting anywhere.
  2. jcsd
  3. Mar 17, 2010 #2


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    The only term that can cause a problem is the last one:


    Since you have set the function equal to 0 at the origin, the following must be true in order to have continuity:

    [tex]\lim_{x,y \rightarrow 0}\frac{4x^6y^2}{(x^4+y^2)^2} = 0[/tex]

    Which part of this fraction has a [itex]4x^4y^2[/itex] in it? That's the logical place to try using the hint.
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