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

In the theory of Lebesgue integration it has been proved that if ff is a real Lipschitz function defined on a segment [a,b]⊂R[a,b]⊂R, then the Newton–Leibniz formula ∫abf′(x)dx=f(b)−f(a) (the fundamental theorem of calculus) holds. This paper extends the fact to the case where the Fréchet derivative f′(⋅)f′(⋅) (which is defined almost everywhere on [a,b][a,b] by the Rademacher theorem) and the Lebesgue integral are replaced, respectively, by the Clarke subdifferential mapping ∂Cf(⋅)∂Cf(⋅) and the Aumann (set-valued) integral. Among other things, we show that f(b)−f(a)∈∫ab∂Cf(x)dx and the equality ∫ab∂Cf(x)dx={f(b)−f(a)} is valid if and only if ff is strictly Hadamard differentiable almost everywhere on [a,b][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.