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Thread: continuous function on R2+

  1. #1
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    continuous function on R2+

    I have a question.
    Show, by using the arithmetic rules, that the function g on R2 + defined by
    g(x,y) = (ln(1 + x2 + y2),ln(1 + x3y))
    is continuous.

    How can I do this? I have no idea how to start.
    Thank you
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  2. #2
    MHF Contributor
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    Re: continuous function on R2+

    \ln{(1+x^3y)} exists only for x^3y > -1 and for t > -1 the logarithm \ln {(1+t)} is continuous. x^3y is continuous everywhere, so \ln{(1+x^3y)} is continuous everywhere it is defined. (What results have I used there?)

    If you can now show that \ln{(1+x^2+y^2)} exists and is continuous on x^3y > -1, you will have that both \ln{(1+x^2+y^2)} and \ln{(1+x^3y)} are continuous. What does that mean for g(x,y)?
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