Let $X$ be a topological space and $f:X\to \mathbb{R}$ and $g:X\to \mathbb{R}$ be continuous functions. How do I show that $h:X\to \mathbb{R}$ where $h:=f+g$ is continuous, would prefer to use the general definition so for and open $U$ in $\mathbb{R}$, $h^-(U)$ is open. Also if $Y$ is another top space and $k:Y\to \mathbb{R}$ is also a continuous function, how do I show $l:X\times Y\to \mathbb{R}$, defined by $l:=f+k$ is also continuous. Any help please. Thanks in advance.
[Math] Continuity of the sum of continuous functions
continuitygeneral-topology
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Best Answer
You should consider the maps
If you can show continuity for all of these maps, you have also shown continuity for $f + g = (+) ∘ (f,g)$ and $f + k = (+) ∘ (f × k)$ (not sure if “$f+k$” isn’t a bit of a misleading notation here).
For the continuity of $(+)$: For any $c ∈ ℝ$, every $ε/2$-maximum-ball around any preimage $(a,b) ∈ (+)^{-1}(c)$ stays within an $ε$-ball of $c$ by the triangle inequality. This implies that preimages of open sets are open. (Essentially: $|(a' + b') - (a + b)| ≤ |a-a'| + |b-b'|$.)