If $f,f':X \rightarrow Y$ ($X,Y$ topological spaces) are continuous functions respectively defined on disjoint closed subsets $A,A'\subset X$ . Then their disjoint union; the function $F:A\cup A'\rightarrow Y$, defined as $f$ on $A$ and $f'$ on $A'$ is continuous. Is the same result true for general sheaves?
That is suppose $X$ is a topological space and $\mathcal{F}$ is a sheaf of $R$-modules over $X$. Let $A,A'\subset X$ be disjoint closed subsets. Then given sections $s\in \mathcal{F}(A)$ and $s'\in \mathcal{F}(A')$ is there a way of associating a section $S\in \mathcal{F}(A\cup A')$?
Is this true in general or do we need specific conditions on the sheaf or the space, if so what are these? If this is true is this association unique?