  This note investigates functions from $\mathbb{R}^d$ to $\mathbb{R} \cup
\{\pm \infty\}$ that satisfy axioms of linearity wherever allowed by
extended-value arithmetic. They have a nontrivial structure defined inductively
on $d$, and unlike finite linear functions, they require $\Omega(d^2)$
parameters to uniquely identify. In particular they can capture vertical
tangent planes to epigraphs: a function (never $-\infty$) is convex if and only
if it has an extended-valued subgradient at every point in its effective
domain, if and only if it is the supremum of a family of "affine extended"
functions. These results are applied to the well-known characterization of
proper scoring rules, for the finite-dimensional case: it is carefully and
rigorously extended here to a more constructive form. In particular it is
investigated when proper scoring rules can be constructed from a given convex
function.
