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, TD 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.


Convex Function Additive Function Measure Zero Simple Condition Density Point 
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© Birkhäuser Verlag AG 2009

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