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Math Help - Continuity of a 2 variables function in R2

  1. #1
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    Continuity of a 2 variables function in R2

    Hello again,

    I have some troubles doing this exercise, it asks me to study the continuity of the following function in R2:

    F(x,y) = < F1(x,y) , F2(x,y) >

    Where those functions are:

    F1 (x,y) = 0, if y = x^2

    F1 (x,y) = (x^2+y) / (x^2-y), if y != x^2


    F2 (x,y) = 0, if x^2+y^2 = 0

    F2 (x,y) = (x^2-y)*sin(1/(x^2+y^2)), if x^2+y^2 != 0


    Now, I suppose that the function F(x,y) is continuos for each point of R2 where both F1 and F2 are continuos. Is that ok?
    Then I ve studied the continuity of those function:

    F2 is continuos for each point of R2 because the only interesting point is (0,0) since this is the unique point where the condition x^2+y^2=0 become true and at this point the function seems to be continuos (using polar coordinates and r ->0).
    F1 is not continuos in (0,0) since by taking 2 different paths (x,0) and (0,y) the limit isn't the same but how can I prove the continuity for all the points standing on y=x^2?

    Thanks in advance!
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  2. #2
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    Take (x_n,y_n) = (a+\tfrac1n,a^2), so that (x_n,y_n)\to(a,a^2) as n\to\infty. Check that F_1(x_n,y_n)\not\to F_1(a,a^2). So the function is not continuous at points on the parabola.
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