In this chapter, we (a) discuss the notion of lower semicontinuity of a function and demonstrate that functions with this property have closed epigraphs, (b) show that the pointwise supremum of a family of lower semicontinous functions is lower discontinuous, (c) demonstrate that a proper lower semiconscious convex function is the pointwise supremum of the affine minorants of the function, (d) introduce the notion of a subgradient and the subdifferential of a convex function at a point and demonstrate existence of subgradients at points from the relative interior of the function’s domain, (e) outline elementary rules of subdifferential calculus, and (f) establish basic properties of the directional derivatives of convex functions and the connection between directional derivatives and subdifferentials.
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