Integral of the Clarke subdifferential mapping and a generalized Newton-Leibniz formula

In the theory of Lebesgue integration it has been proved that if f is a real Lipschitz function defined on a segment [a,b]⊂R, then the Newton-Leibniz formula (the fundamental theorem of calculus) holds. This paper extends the fact to the case where the Fréchet derivative f^′(⋅) (which is defined almost everywhere on [a,b] by the Rademacher theorem) and the Lebesgue integral are replaced, respectively, by the Clarke subdifferential mapping ∂_Cf(⋅) and the Aumann (set-valued) integral. Among other things, we show that and the equality is valid if and only if f is strictly Hadamard differentiable almost everywhere on [a,b]. The result is derived from a general representation formula, which we obtain herein for the integral of the Clarke subdifferential mapping of a Lipschitz function defined on a separable Banach space.