Chain differentials with an application to the mathematical fear operator

We introduce a concept of derivative in a topological vector space that yields the chain rule of differentiation for composition of functions, akin to, but simpler than, the epiderivative of Aubin-Frankowska. We show that if a function has a Gateaux differential with suitable continuity properties, it is a chain differential. As an example, we show that the mathematical fear operator is chain-differentiable with respect to the cost density distribution in the space of continuous functions endowed with the topology of pointwise convergence uniform on every compact subsets. (c) 2005 Elsevier Ltd. All rights reserved.