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If $f,g : X \to \mathbb{R}$ are continuous, prove the following function is also continuous from $X$ to $\mathbb{R}$.

$f + g$ defined by $(f + g)(x) = f(x) + g(x)$.

This is for topology, so my first thought is to show that the inverse image is open. However, I didn't know if you could show this without using any epsilons or deltas.

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You would have to use the $\,\delta\,$ in $\,X\,$ , which may prove to be impossible for general topological spaces. –  DonAntonio Feb 25 '13 at 3:02
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Hint: Show that $+\colon\Bbb{R\times R\to R}$ defined by $+(x,y)=x+y$ is continuous and that $h\colon X\to\Bbb{R\times R}$ defined by $h(x)=\langle f(x),g(x)\rangle$ is continuous.

Then use the fact/show that the composition of two continuous functions is continuous.

Remember: It suffices to show that basic open sets have an open preimage in order to conclude that the function is open. So it suffices to show for intervals with $\Bbb R$, and either squares or open balls for $\Bbb{R\times R}$.

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