Topology of the Real Line Continuity and Compactness Limit Superior and Limit Inferior of Functions Lower Semicontinuity and Upper Semicontinuity Convex Functions and Derivatives Nondifferentiable Convex Functions and Subdifferentials 5. ADDITIONAL TOPICS In this chapter we introduce topological properties of the real line and provide generalizations of the main results about continuous functions in this context. In addition, we introduce the notion of upper and lower semicontinuous functions and give further generalizations of the extreme value theorem for them. Finally, we discuss convex functions and study the existence of extrema for them using the notion of subdifferential. 5.1 Topology of the Real Line In this section we introduce some topological concepts on the real line R. These concepts capture the essential characteristics of sets that are relevant for the study of continuity. In particular they will allow extensions of the extreme value theorem and other results to the case when the domain of the function may not be an interval. The open ball inRwith center a∈Rand radius δ >0 is the set B(a;δ) = (a−δ,a+δ). Definition 5.1.1 A subset Aof Ris said to be open if for each a∈A, there exists δ >0 such that B(a;δ) ⊂A. ■ Example 5.1.1 The following examples illustrate the concept of open set. (a) Any open interval A= (c,d) is open. Indeed, for eacha∈A, one has c <a<d. Let δ =min{a−c,d−a}. Then B(a;δ) = (a−δ,a+δ) ⊂A. Therefore, Ais open.
RkJQdWJsaXNoZXIy NTc4NTAz