Mean-value theorem with small subdifferentials

We prove a mean-value theorem for lower semicontinuous functions on a large class of Banach spaces which contains the class of Asplund spaces, in particular reflexive Banach spaces and Banach spaces with a separable dual. It involves the lower subdifferential (or contingent subdifferential) and the Frechet sub differentials, which are among the smallest sub differentials known to date. It follows that the estimates which it provides require weak assumptions and are accurate. When the function is locally Lipschitzian, we get a simple statement which refines the Lebourg mean-value theorem.