143 Figure 5.5: Defnition of subderivative. Proof: By Lemma 5.6.1, the slope function φx0 defned by (5.9) is increasing on the interval (x0,∞) and bounded below by φx0 (x0 − 1). By Theorem 3.2.4, the limit f (x) − f (x0) lim φx0 (x)= lim + x→x+ x − x0 0 x→x0 exists and is fnite. Moreover, lim φx0 (x)= inf φx0 (x). + x>x0 x→x 0 ′ Thus, f (x0) exists and + ′ f+(x0)= inf φx0 (x). x>x0 ′ Similarly, f − (x0) exists and ′ f − (x0)= sup φx0 (x). x<x0 Applying Lemma 5.6.1 again, we see that (y) whenever x < x0 < y. φx0 (x) ≤ φx0 ′ ′ This implies f − (x0) ≤ f (x0). The proof is complete. □ + Theorem 5.6.3 Let f : R → R be a convex function and let x0 ∈ R. Then ′ ′ ∂ f (x0)=[ f − (x0), f (x0)]. (5.10) + Proof: Suppose u ∈ ∂ f (x0). By the defnition (5.8), we have u· (x − x0) ≤ f (x) − f (x0) for all x > x0. This implies f (x) − f (x0) u ≤ for all x > x0. x − x0
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