Extended Real-Valued Functions

Abstract

In this chapter, we present basic notations and properties for functions which attain values in $\mathbb {R}\cup \{-\infty ,+\infty \}$, the extended set of real numbers. Of course, such functions may also be real-valued. In the first two sections, we give a short overview about the extended set of real numbers and explain the way in which extended real-valued functions are handled in convex analysis. We also discuss problems that arise by admitting values $\pm \infty$. Afterwards, statements about the continuity and the semicontinuity of extended real-valued functions are summarized. We investigate convexity, generalized convexity, linearity, sublinearity and related properties of the functionals. One section is devoted to monotonicity. Furthermore, basic notions for the variational analysis of extended real-valued functions are collected. Since extended real-valued functions belong to the standard in convex analysis, most of the results in this chapter are well known. We add several new statements in order to point out that some properties of the functions which we focus on in this book turn out to be properties of each extended real-valued function.