Let $X$ be a scheme. $U$ is an open subscheme of $X$. Assume $f$ is a global section on $X$ which is not a zero divisor, then the restriction of $f$ to $U$ is still a non-zero divisor?

If $X$ is affine, the answer is obvious true. I don't know the answer for a general scheme.

This is a question raised in the definition of sheaf of total fraction rings. Some author claim
    U|-> total fraction ring of sections over U
is a presheaf, but I can't see the reason.