If $f$ and $g$ are continuous in $x_0$ prove that $f+g$ is continuous in $x_0$

Question

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.

Answer 1

For $\epsilon>0$

There is an open set $U_1$ with $x_0\in U_1$ such that $||f(x)-f(x_0)||<\epsilon/2$

There is an open set $U_2$ with $x_0\in U_2$ such that $||g(x)-g(x_0)||<\epsilon/2$

So for the open set $U=U_1\cap U_2$ we have $x_0\in U$ and

$||(f+g)(x)-(f+g)f(x_0)||\leq ||f(x)-f(x_0)||+||g(x)-g(x_0)||<\epsilon/2+\epsilon/2=\epsilon$

Which was to be proved.