ABSTRACT
This chapter expands the treatment of functions a bit and looks at the class of lower semicontinuous functions and the class of convex functions. These functions gives us some new insights into how we can try to find extreme values of functions even when there is no compactness. The chapter discusses the smallest and largest values of functions without traditional notions of continuity and compactness. It reviews lower semicontinuous functions and describes extreme values of a continuous function with growth conditions at infinity. The chapter provides a set of examples of lower semicontinuity, and presents extreme values for lower semicontinuous functions with growth conditions at infinity. Convex functions, three points Lemma and the criterion of increasing slopes are described. Left hand and right hand secant slopes are provided with finite supremum and infimum. The chapter reviews the subdifferential of a convex function.
