I have a question. I have to show, by using the arithmetic rules, that the function $g$ on $\Bbb R^+\times\Bbb R^+$ defined by $$g(x,y) = \big(\ln(1+x^2+y^2), \ln(1+x^3y)\big)$$ is continuous.
How must I do that?
Continuous function in $\Bbb R^+\times\Bbb R^+$
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general-topology
1 Answers
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Let $g_1(x,y)=\ln(1+x^2+y^2)$ and $g_2(x,y)= \ln (1+x^3y)$ for $x,y >0$
Then: $g$ is continuous iff $g_1$ and $g_2$ are continuous.
$\ln$ is continuous and the fuctions $1+x^2+y^2$ and $1+x^3y$ are continuous, hence $g_1$ and $g_2$ are continuous, therefore $g$ is continuous.
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0Thank you, but how do you prove that the function ( 1+x^2 +y^2 and 1+x^3y) are continuous? – 2017-01-31