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December 19th, 2014, 05:54 PM   #1
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How to show the map $f:\mathbb R^2\rightarrow\mathbb R$?

How to show the map $f:\mathbb R^2\rightarrow\mathbb R$, defined as $f(x,y)=x+y$ is continuous for all $(x,y)\in\mathbb R^2$?

Question: I want to show the map $f:\mathbb R^2\rightarrow\mathbb R$, defined as $f(x,y)=x+y$ is continuous for all $(x,y)\in\mathbb R^2$.

Issue: I know how to prove this via the epsilon-delta way. I want to prove this using the projection functions $p_1,p_2: \mathbb R^2\rightarrow\mathbb R$ where $p_1$ maps $(x,y)\rightarrow x$, similarly for $p_2$. Now my books says via a result based on continuous functions from $\mathbb R\rightarrow\mathbb R$ that the sum of $p_1+p_2$ is continuous but I don't see how that would be possible as the domain of the functions are $\mathbb R$ and not $\mathbb R^2$.
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