Morphisms and Whitney Sum of bundles

Question

Suppose that $X$ is a topological space and $(E,p_1)$ and $(F, p_2)$ are locally trivial vector bundles over $X$. Consider bundle $E \oplus F$. Is it true that $i_E$ defined by $E_x \to (E \oplus F)_x$ and $Pr_E$ defined by $(E \oplus F)_x \to E_x$ are morphisms of vector bundles, i.e. are continous?

Answer 1

Yes, they are. It should be clear that this is true for the trivial bundles, so you can prove the general case by noting that it is a local issue, and then trivializing over the base.