Consider the pre-sheaf, F, of constant functions with values in some set over I(the interval). Consider the open set V = [0,a) U (b,1] , where a<b

Then consider the open cover {A,B} = {[0,a), (b,1]} then the intersection is empty, but there is no isomorphism between F(V) and F(A)*F(B), where

I mean elements of F(A) and F(B) that agree on their intersection.

Hence F is not a sheaf and the sheaf associated with F is the sheaf of locally constant functions. Is this correct because I have read in books that F is a sheaf; but were they just saving on ink by not writing locally?