In Karoubi's “K-Theory”, Theorem I.3.1, he states that given real or complex vector bundles $(E,π), (E',π')$ over $X$, any family of morphisms $\alpha_i\colon E\vert_{U_i}\to E'\vert_{U_i}$ agreeing on intersections uniquely extends to a global morphism $E\to E'$. This sounded an awful lot like the gluing axiom of sheaves. I read up to formulate the following:
Observation If $\mathscr O_X$ is the ringed space of continuous functions $X\to \mathbb R\text{ or }\mathbb C$, then the continuous sections of a vector bundle $E$ give rise to a sheaf $\Gamma(E)$ of $\mathscr O_X$-modules.
I also understood that since trivializations are given by n linearly independent sections, a local trivialization over $U$ can be identified with an isomorphism $Γ(E)\vert_U\to \mathscr O_U{}^n$ of sheaves of $\mathscr O_U$-modules. Consequently, local trivializability should translate to $Γ$ being locally free.
My intuition would be that a morphism of Vector bundles is uniquely determined by the corresponding morphism on the sections, since the latter just transports more than a single vector at the same time. However, I have trouble proving the following:
Conjecture Let $A\colon Γ(E)\to Γ(E')$ be a morphism of sheaves of $\mathscr O_X$-modules. Then there is a unique morphism $\alpha\colon E\to E'$ of vector bundles such that for every $x\in X$ and section $s\in Γ(E)$, we have $α(s(x)) = As(x)$.
Since every $e\in E$ in the fiber of $x$ admits a section $s$ with $s(x)=e$, it is clear that $α$ is completely determined by $A$. However, how do we know that $α$ is continuous?
I hope I didn't make any wrong claims, I'm not too strong with sheaf theory yet.