3
$\begingroup$

Show that $f(x,y)=2x-y$ is uniformly continuous in $\mathbb{R^2}$. Use the definition. How can I do this using just the definition of uniform continuity?

  • 0
    The title should be: "Show that f(x,y) = 2x - y is uniformly continuous"2012-12-20

1 Answers 1

4

Let $\epsilon>0$ and $x,y,a,b\in \mathbb{R}$. We want $\left|f(x,y)-f(a,b)\right|<\epsilon\implies \left|2x-y-2a+b\right|<\epsilon$ Because $\left|2x-y-2a+b\right|\le 2\left|x-a\right|+\left|y-b\right|$ it suffices $2\left|x-a\right|+\left|y-b\right|<\epsilon$ when $\left|(x-a,y-b)\right|<\delta\implies\left|x-a\right|<\delta\text{ and }\left|y-b\right|<\delta$ Choosing $\delta=\frac{\epsilon}{3}>0$ will do the trick. Because $\delta$ doesn't depend on $x,y,a,b$, the continuity is uniform