Introduction to Mathematical Analysis I - Second Edition

133 Figure 4.7: A nondifferential convex function. Example 4.7.1 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 4.7.1 Let f : R → R be a convex function. Fix a ∈ R . Define the slope function φ a by φ a ( x ) = f ( x ) − f ( a ) x − a (4.17) for x ∈ ( − ∞ , a ) ∪ ( a , ∞ ) . Then, for x 1 , x 2 ∈ ( − ∞ , a ) ∪ ( a , ∞ ) with x 1 < x 2 , we have φ a ( x 1 ) ≤ φ a ( x 2 ) . Proof: This lemma follows directly from Lemma 4.6.5 . Theorem 4.7.2 Let f : R → R be a convex function and let ¯ x ∈ R . Then f has left derivative and right derivative at ¯ x . Moreover, sup x < ¯ x φ ¯ x ( x ) = f 0 − ( ¯ x ) ≤ f 0 + ( ¯ x ) = inf x > ¯ x φ ¯ x ( x ) , where φ ¯ x is defined in ( 4.17 ) .