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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?

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    Can you use the definition of continuity? Can you show that in that case, given an $\epsilon>0$, the same $\delta$ works everywhere? That is what uniform continuity is, so you're done!2012-12-20
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    The title should be: "Show that f(x,y) = 2x - y is uniformly continuous"2012-12-20

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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