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Math Help - Proving where a multivariable function is continuous

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    Senior Member Pinkk's Avatar
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    Proving where a multivariable function is continuous

    Let f(x,y) = y(y-x^{2})/x^{4} if 0 < y < x^{2} and f(x,y) = 0 otherwise. Prove that f is discontinuous only at (0,0).

    So I am fairly certain I know how to show the function is discontinuous at (0,0):

    If f was continuous at (0,0), then \lim_{(x,y)\to (0,0)} f(x,y) = f(0,0) = 0. However, if we approach (0,0) along the parabola y= 2x^{2}, then we have that \lim_{(x, 2x^{2})\to (0,0)} 2x^{2}(2x^{2} - x^{2})/x^{4} = \lim_{(x, 2x^{2})\to (0,0)} 2 = 2 since 0 < 2x^{2} < x^{2} as (x, 2x^{2}) gets sufficiently closer and closer to the origin. Therefore, f is discontinuous at (0,0).

    Is that proof correct, and if not, what is wrong?

    Moving to the next portion, I am stuck; how do I prove that for all other points, f is continuous. I have a general idea that I need to check points outside the region 0 < y < x^{2}, inside that region, and then on the boundaries of the region (namely points along y= x^{2} and y=0. I am having trouble formalizing this into a proof though. Thanks for any help.

    Edit: Er, stupid me, meant \frac{1}{2}x^{2}, not 2x^{2}, otherwise what I wrote doesn't make sense.
    Last edited by Pinkk; August 31st 2010 at 05:20 PM.
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