528 APPENDIX B Finally, we record the relation between the concepts open and closed. These concepts are complementary: a subset A R n is open precisely if the complement R \ A is closed. B.3 CONVEXITY We give the basic definitions and properties of convexity. A set C in R n is called a convex set if it contains for each two of its points all intermediate points: a, b C, 0 1 (1 - )a + b C. A function f : A R defined on a set A R n is called a convex function if its epigraph {(a, ) : a A, f (a)} is a convex set; this implies that A is a convex set. This definition is equivalent to the usual one. Theorem B.4 A function f : A R defined on a set A R is convex precisely if the set A is convex, and, moreover, the following inequality holds true: f ((1 - )a + b) (1 - )f (a) + f (b) for all a, b A and 0 1 The continuity of convex functions is intuitively obvious. However, at boundary points discontinuity is possible. Theorem B.5 A convex function f : A R is continuous at all interior points of its domain A. The qualification "interior" cannot be omitted here, as the following example illustrates. Example B.3.1 unit sphere Take an arbitrary function g : S n-1 R + on the S n-1 = {x R n : |x| = 1}. Extend it to a function f on the unit ball B n = {x R n : |x| 1}