Geodesic
Notes on Extended Real- and Set-Valued Functions
Journal of convex analysis, Tome 19 (2012) no. 2, pp. 355-384.

Voir la notice de l'article provenant de la source Heldermann Verlag

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.
Classification : 49N15, 54C60, 90C46
Mots-clés : Extended real-valued functions, directional derivative, subdifferential, Fenchel conjugate, set-valued function, conlinear space, infimal convolution 
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     title = {Notes on {Extended} {Real-} and {Set-Valued} {Functions}},
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A. H. Hamel; C. Schrage. Notes on Extended Real- and Set-Valued Functions. Journal of convex analysis, Tome 19 (2012) no. 2, pp. 355-384. http://geodesic.mathdoc.fr/item/JCA_2012_19_2_JCA_2012_19_2_a2/