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Math Help - Uniform Continuity

  1. #1
    Senior Member roninpro's Avatar
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    Uniform Continuity

    Hello, I have been asked to show that the function

    f(x,y)=\ln(1+x^2+y^2)

    is uniformly continuous on \mathbb{R}^2.

    I played with it quite a bit and it is a complete mess. Any suggestions would be appreciated.
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  2. #2
    Member Focus's Avatar
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    Quote Originally Posted by roninpro View Post
    Hello, I have been asked to show that the function

    f(x,y)=\ln(1+x^2+y^2)

    is uniformly continuous on \mathbb{R}^2.

    I played with it quite a bit and it is a complete mess. Any suggestions would be appreciated.
    Show that g(x,y)=1+x^2+y^2 is continuous between \mathbb{R}^2 \rightarrow [1,\infty). And prove that \ln:[1,\infty)\rightarrow \mathbb{R} is continuous.
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  3. #3
    Senior Member roninpro's Avatar
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    Thank you for the reply.

    Although I can see that proving that g(x,y) and \ln(x) are both continuous implies that \ln(g(x,y)) is continuous, I do not see why it necessarily implies that it is uniformly continuous.
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