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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+

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

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