How to show that $f(x,y) = |x| + |y|$ is continous at origin

Question

How to show that $f(x,y) = |x| + |y|$ is continous at origin. CLearly it goes to 0 , but how do i prove it? Thanks

Answer 1

Since $|x|,|y| \le \sqrt{x^2+y^2}$ we have: $|x|+|y|\le 2 \sqrt{x^2+y^2} \rightarrow 0 $ as $(x,y)\rightarrow 0$.

Answer 2

If you know that all norms are equivalent on real finite dimensional vectors spaces (in particular $\mathbb R^2$), then you can just say that $f(x,y)$ which is a norm is continuous.