Different extensions of the notion of gradient in differentiable analysis and of subdifferential in convex analysis are used in order to derive mean value theorems for non-differentiable functions. Previous results dealing with convex or locally Lipschitz real-valued functions are extended. In addition, mean value theorems in an equality form are described for locally Lipschitz vector-valued mappings.
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J-B. Hiriart-Urruty (1980) studied this question.
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