Introduction to Mathematical Analysis I - Second Edition

134 4.7 NONDIFFERENTIABLE CONVEX FUNCTIONS AND SUBDIFFERENTIALS Figure 4.8: Definition of subderivative. Proof: By Lemma 4.7.1 , the slope function φ ¯ x defined by ( 4.17 ) is increasing on the interval ( ¯ x , ∞ ) and bounded below by φ ¯ x ( ¯ x − 1 ) . By Theorem 3.2.4 , the limit lim x → ¯ x + φ ¯ x ( x ) = lim x → ¯ x + f ( x ) − f ( ¯ x ) x − ¯ x exists and is finite. Moreover, lim x → ¯ x + φ ¯ x ( x ) = inf x > ¯ x φ ¯ x ( x ) . Thus, f 0 + ( ¯ x ) exists and f 0 + ( ¯ x ) = inf x > ¯ x φ ¯ x ( x ) . Similarly, f 0− ( ¯ x ) exists and f 0 − ( ¯ x ) = sup x < ¯ x φ ¯ x ( x ) . Applying Lemma 4.7.1 again, we see that φ ¯ x ( x ) ≤ φ ¯ x ( y ) whenever x < ¯ x < y . This implies f 0− ( ¯ x ) ≤ f 0 + ( ¯ x ) . The proof is complete. Theorem 4.7.3 Let f : R → R be a convex function and let ¯ x ∈ R . Then ∂ f ( ¯ x ) = [ f 0 − ( ¯ x ) , f 0 + ( ¯ x )] . (4.18) Proof: Suppose u ∈ ∂ f ( ¯ x ) . By the definition ( 4.16 ) , we have u · ( x − ¯ x ) ≤ f ( x ) − f ( ¯ x ) for all x > ¯ x . This implies u ≤ f ( x ) − f ( ¯ x ) x − ¯ x for all x > ¯ x .