At http://en.wikipedia.org/wiki/Vector_bundle we have:
"Given a vector bundle $\pi : E \rightarrow X$ and an open subset $U$ of $X$, we can consider sections of $\pi$ on $U$, i.e. continuous functions $s : U \rightarrow E$ where the composite $\pi \circ s$ is such that $(π \circ s)(u) = u$ for all $u$ in $U$."
And:
"Let $F(U)$ be the set of all sections on $U$. $F(U)$ always contains at least one element, namely the zero section: the function s that maps every element $x$ of $U$ to the zero element of the vector space $\pi^{-1}(\{x\})$. With the pointwise addition and scalar multiplication of sections, $F(U)$ becomes itself a real vector space."
I'm having trouble showing that the pointwise addition of two sections gives a continuous map, I can prove that the addition and scalar multiplication in any one given fibre is continuous (in the topology inherited from $E$) and I have tried without success using these continuities on individual fibres, I have also tried using the local trivialisations, again without success. After some time of failure I quickly had a try at showing that the scalar multiples of sections are also continuous and seem to have run into similar difficulties.
Some detailed guidance would be much appreciated.
Best Answer
Let $x \in X$ and $U$ be a neighborhood of $x$ homeomorphic to ${\bf R}^n$. We want to prove that addition of two sections is continuous at $x$. Thanks to the local trivialization of $E$ over $U$, we can equivalently talk about the product bundle $\pi: U \times {\bf R^n} \to U$. A section of this bundle is given by a map $s : U \to U \times {\bf R^n}$ such that $\pi (s (y)) = y$ for all $y \in U$. So such a section is nothing more than a map $s : U \to {\bf R}^n$. There the additivity of sections should be obvious.
I'll leave it up to you to fill in the details by following all the isomorphisms I used. You also need to verify that the answer doesn't depend on the choice of trivialization (which follows from the cocycle conditions).