148 5.6 Nondifferentiable Convex Functions and Subdifferentials Similarly, if a1 <a2 <· · · <a2k and g(x) = 2k ∑ i=1| x−ai|. Then 0 ∈∂g(x0) if and only if x0 ∈[ak,ak+1]. Thus, g has an absolute minimum at any point of [ak,ak+1]. The following theorem is a version of the Mean Value Theorem (Theorem 4.2.3) for nondifferentiable functions. Figure 5.9: Subdifferential mean value theorem. Theorem 5.6.8 Let f : R→Rbe a convex function and let a<b. Then there exists c ∈(a,b) such that f (b)−f (a) b−a ∈ ∂ f (c). (5.11) Proof: Define g(x) =f (x)− f (b)−f (a) b−a (x−a)+f (a) . Thengis a convex function andg(a) =g(b). Thus, ghas a local minimum at some c ∈(a,b) and, hence, galso has an absolute minimum at c. Observe that the function h(x) =− f (b)−f (a) b−a (x−a)+f (a)
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