Introduction to Mathematical Analysis I - Second Edition

139 The following theorem is a version of the Mean Value Theorem (Theorem 4.2.3 ) for nondifferen- tiable functions. Figure 4.12: Subdifferential mean value theorem. Theorem 4.7.9 Let f : R → R be a convex function and let a < b . Then there exists c ∈ ( a , b ) such that f ( b ) − f ( a ) b − a ∈ ∂ f ( c ) . (4.19) Proof: Define g ( x ) = f ( x ) − f ( b ) − f ( a ) b − a ( x − a )+ f ( a ) . Then g is a convex function and g ( a ) = g ( b ) . Thus, g has a local minimum at some c ∈ ( a , b ) and, hence, g also has an absolute minimum at c . Observe that the function h ( x ) = − f ( b ) − f ( a ) b − a ( x − a )+ f ( a ) is differentiable at c and, hence, ∂ h ( c ) = { h 0 ( c ) } = − f ( b ) − f ( a ) b − a .