143 Figure 5.5: Definition of subderivative. Proof: By Lemma 5.6.1, the slope function φx0 defined by (5.9) is increasing on the interval (x0,∞) and bounded below byφx0(x0 −1). By Theorem 3.2.4, the limit lim x→x+ 0 φx0(x)= lim x→x+ 0 f (x)−f (x0) x−x0 exists and is finite. Moreover, lim x→x+ 0 φx0(x)= inf x>x0 φx0(x). Thus, f ′+(x0) exists and f ′+(x0)= inf x>x0 φx0(x). Similarly, f ′ − (x0) exists and f ′ − (x0)=sup x<x0 φx0(x). Applying Lemma 5.6.1 again, we see that φx0(x) ≤φx0(y) whenever x <x0 <y. This implies f ′ − (x0) ≤f ′+(x0). The proof is complete. □ Theorem 5.6.3 Let f : R→Rbe a convex function and let x0 ∈R. Then ∂ f (x0)=[ f ′ − (x0), f ′+(x0)]. (5.10) Proof: Suppose u∈∂ f (x0). By the definition (5.8), we have u·(x−x0) ≤f (x)−f (x0) for all x >x0. This implies u≤ f (x)−f (x0) x−x0 for all x >x0.
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