Limiting subdifferentials of indefinite integrals

We compute the limiting subdifferential ∂F( ¯ x) of the indefinite integral of the form F(x) = x a f (t)dt where f is an essentially bounded measurable function, or a function continuous on an interval containing ¯x ∈ R (except for, possibly, ¯x ), or a step-function which has a countable number of steps around ¯x . The related problem of computing the Aumann integral of the limiting subdifferential mapping ∂f (·), where f is a Lipschitz real function defined on an open set U ⊂ Rn, is also investigated. © 2007 Elsevier Inc. All rights reserved.