I want to determine if the function $f:\mathbb R^2 \to \mathbb R^2$ defined as $f(x,y)=(2x+y,x+2y) $ is Lipschitz continuous.
I know that you can consider coordinate functions $f_1,f_2:\mathbb R^2 \to \mathbb R$
$$f_1(x,y)=2x+y \text{ and } f_2(x,y)=x+2y $$ and if I could prove that they are LC (or not), I would know that $f$ is LC (or not).
Problem is, using the definition it gets tedious and really complicated. Not sure if I could use another metric ($1$ or $\infty$) becuase $2$-metric is awful.
Any help would be appreciated.
It can be calculated directly, if you want: First of all note that $\|(x,y)\| = \|(y,x)\|$. For $z := (x,y) \in \mathbb R^2$ let's denote $\overline z := (y,x)$. For such a $z$ you can see that
$$f(z) = 2z + \overline z.$$
Now we compute for $x_1, x_2 \in \mathbb R^2$ that
$$\| f(x_1) - f(x_2) \| = \| 2x_1 - 2x_2 + \overline{x_1} - \overline{x_2} \| = \|2(x_1 - x_2) + \overline{(x_1 - x_2)}\| \\ \leq 2 \|x_1 - x_2\| + \|\overline{x_1 - x_2}\| = 3 \|x_1 - x_2 \|.$$