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 Gâteaux 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.