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Uniform continuity of sum function

Posted November 21st, 2007 by Structure

Let f(x,y)=x+y
This function is uniform continuous, as it is easy to verify using "Continuous Function on metric space"previous criteria.
Let $ (a,b)\in R^2 $ and let $ \epsilon >0 $ for all $ (x,y)\in R^2 $ such as $ \sqrt{(x-a)^2+(y-b)^2}<\delta $ we have
$ |x+y-a-b|\leq|x-a|+|y-b|\le\sqrt{2}\sqrt{(x-a)^2+(y-b)^2}<\sqrt{2}\delta $ and the last product is less or equal to $ \epsilon $ if $ \sqrt{2}\delta\leq\epsilon $ So we can chose $ \delta=\frac{\epsilon}{\sqrt{2}} $
As the choice of $ \delta $ does not depend on $ (a,b)\in R^2 $ the function is uniform continuous on $ R^2 $

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