Notes on Extended Real- and Set-Valued Functions

An order theoretic and algebraic framework for the extended real numbers is established which includes extensions of the usual difference to expressions involving -∞ and/or +∞, so-called residuations. New 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.