In Hartshorne Algebraic Geometry the definition of the sheaf of rings attached to the spectrum of a ring $A$ to define an affine scheme has the following:
Next we define the sheaf of rings $O_X$ on $X:=Spec(A)$. For each prime ideal $p \in X$ we define $O_X U$ to be the set of functions $s:U \rightarrow \amalg_{p \in U} A_p$ such that $sp \in A_p$ for each $p$ and such that $s$ is locally a quotient of elements of $A$:
to be precise we require for each $p \in U$, an open set $V$ contained in $U$ and containing $p$; and elements $a,b \in A$ such that for each $q \in V$, $b \notin q$ and $sq=a/b$ in $A_q$.
(in chapter II, section 2; just after lemma 2.1; and page 70 in my edition):
(Such an $s$ is otherwise known as a regular function; and presumably Hartshorne chose this letter as it is a section as the definition above makes clear). This makes it look like that $s$ is locally constant; is this right? A 'sloping straight line', for example, isn't locally constant; should it say instead:
for each $q \in V$ there is $a,b \in A$ such that $b \notin q$ and $sq=a/b$ in $A_q$?
Hartshorne's statement is correct as written. I would have thought that after all your questions about structure sheaves and etale spaces, you would recognize that this is the "etale space" point-of-view on the structure sheaf: a section of a sheaf consists of given an element of the stalk at each point, chosen so that these elements are locally constant.