Continuous Functions

Abstract

In this book we use semicontinuous functions when providing sufficient conditions for a function to achieve its maximum on a compact set. They represent a broader class of functions than continuous functions but still possess many of the same properties. We state results in considerable generality, although in applications we usually consider real-valued functions on Euclidean space. Royden (1963) provides a nice mathematical background for material in this appendix. References in the Markov decision process literature include papers of Maitra (1968) and Himmelberg, Parthasarathy and van Vleck (19761, and the books of Hinderer (1970, pp. 31-35 and 113-1171, Bertsekas and Shreve (1978), Dynkin and Yushkevich (19791, and Hernandez-Lerma (1989). Our presentation follows Maitra and Hinderer.