Boundedness and Continuity of Convex Functions and Additive Functions
The Theorem of Bernstein-Doetsch (cf., in particular, Corollary 6.4.1) says that if D ⊂ ℝN is a convex and open set, f: D → ℝ is a convex function, T ⊂ D is open and non-empty, and f is bounded above on T, then f is continuous in D. Are there other sets T with this property? What are possibly weak conditions which assure the continuity of a convex function, or of an additive function? In this and in the next chapter we will deal with such questions.
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