Lower semicontinuous convex functions

Abstract

Our differentiability results for convex functions made heavy and consistent use of continuity, but in both the theoretical and applied aspects of convex functions, it is sometimes desirable to weaken this hypothesis. Lower semicontinuity is precisely what is needed. Uncomfortable as it may seem at first, the subject is best treated by introducing the seeming complication of admitting extended real-valued functions, that is, functions with values in R ∪ {00}. We adopt the conventions that

$$ r \cdot \infty = \infty and \left( { - r} \right) \cdot \infty = - \infty if r > 0, and r \pm \infty for all r \in R. $$

(We won’t have occasion to worry about ∞ — ∞ or 0 · ∞.)