Sorry, I meant the claim about f and g. Assuming you meant that F(I) should be continuous functions, you can construct an h from f,g to be cont on I and J, no? So it's not an axiom. Just making sure I understand correctly...
Again, that was just an example. F is just one possible sheaf on the real line, and in the case of F, yes, continuous functions can be stitched together.
You could define A(I) = { } for a trivial example of a different sheaf A where the "data" (always an empty set, regardless of I) is very different from what it was in the case of F (the set of continuous functions on I).
