  An order theoretic and algebraic framework for the extended real numbers is
established which includes extensions of the usual difference to expressions
involving $-\infty$ and/or $+\infty$, so-called residuations. Based on this,
definitions and results for directional derivatives, subdifferentials and
Legendre--Fenchel conjugates for extended real-valued functions are given which
admit to include the proper as well as the improper case. For set-valued
functions, scalar representation theorems and a new conjugation theory are
established. The common denominator is that the appropriate image spaces for
set-valued functions share fundamental structures with the extended real
numbers: They are order complete, residuated monoids with a multiplication by
non-negative real numbers.
