How to solve this problem?

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

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

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