Im getting trouble trying to formulate and making this proof. The exercise is stated like this:
Let $f:A\to F$ and $g:B\to F$ with $A,B\subseteq X$ where $F$ is a normed space and $X$ is a topological space. Prove that if $f$ and $g$ are continuous at $x_0$ then $f+g$ is continuous at $x_0$.
The proof is easy if $X$ would be a metric space instead of an arbitrary topological space.
The problem here is that for arbitrary topological spaces I cant use a sequential characterization of continuity.
Then I tried to play around with the topological definition of continuity and the algebra of open balls in normed spaces but I dont get any clue.
Some hint will be appreciated, thank you.