  We investigate finite sets of rational functions $\{ f_{1},f_{2}, \dots,
f_{r} \}$ defined over some number field $K$ satisfying that any $t_{0} \in K$
is a $K_{p}$-value of one of the functions $f_{i}$ for almost all primes $p$ of
$K$. We give strong necessary conditions on the shape of functions appearing in
a minimal set with this property, as well as numerous concrete examples showing
that these necessary conditions are in a way also close to sufficient. We
connect the problem to well-studied concepts such as intersective polynomials
and arithmetically exceptional functions.
