Introduction to Mathematical Analysis I - 3rd Edition

142 5.6 Nondifferentiable Convex Functions and Subdifferentials Figure 5.4: A nondifferential convex function. ■ Example 5.6.1 Let f (x) =|x|. Then ∂ f (0) = [−1,1]. Indeed, for anyu∈∂ f (0), we have u· x =u(x−0) ≤f (x)−f (0) =|x| for all x ∈R. In particular, u· 1≤ |1| =1 and u· (−1) =−u≤ | −1| =1. Thus, u∈[−1,1]. It follows that ∂ f (0) ⊂[−1,1]. For any u∈[−1,1], we have |u| ≤1. Then u· x ≤ |u· x| =|u||x| ≤ |x| for all x ∈R. This implies u∈∂ f (0). Therefore, ∂ f (0) = [−1,1]. Lemma 5.6.1 Let f : R→Rbe a convex function. Fixa∈R. Define the slope functionφa by φa(x) = f (x)−f (a) x−a (5.9) for x ∈(−∞,a)∪(a,∞). Then, for x1,x2 ∈(−∞,a)∪(a,∞) withx1 <x2, we have φa(x1) ≤φa(x2). Proof: This lemma follows directly from Lemma 5.5.5. □ Theorem 5.6.2 Let f : R→Rbe a convex function and let x0 ∈R. Then f has a left derivative and a right derivative at x0. Moreover, sup x<x0 φx0(x) =f ′ −(x0) ≤f ′ +(x0) = inf x>x0 φx0(x), where φx0 is defined in (5.9).

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