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
$\displaystyle \ln{(1+x^3y)}$ 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 $\displaystyle g(x,y)$?