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 any u ∈ ∂ 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 → R be a convex function. Fix a ∈ R. Defne the slope function φa by f (x) − f (a) φa(x)= (5.9) x − a for x ∈ (−∞,a) ∪ (a,∞). Then, for x1,x2 ∈ (−∞,a) ∪ (a,∞) with x1 < x2, we have φa(x1) ≤ φa(x2). Proof: This lemma follows directly from Lemma 5.5.5. □ Theorem 5.6.2 Let f : R → R be a convex function and let x0 ∈ R. Then f has a left derivative and a right derivative at x0. Moreover, ′ ′ sup φx0 (x)= f − (x0) ≤ f (x0)= inf φx0 (x), + x>x0 x<x 0 where φx0 is defned in (5.9).
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