1. ## Uniform Continuity

Hello, I have been asked to show that the function

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

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

I played with it quite a bit and it is a complete mess. Any suggestions would be appreciated.

2. Originally Posted by roninpro
Hello, I have been asked to show that the function

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

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

I played with it quite a bit and it is a complete mess. Any suggestions would be appreciated.
Show that $\displaystyle g(x,y)=1+x^2+y^2$ is continuous between $\displaystyle \mathbb{R}^2 \rightarrow [1,\infty)$. And prove that $\displaystyle \ln:[1,\infty)\rightarrow \mathbb{R}$ is continuous.

3. Thank you for the reply.

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