Directional upper derivatives and the chain rule formula for locally Lipschitz functions on Banach spaces

Motivated by an attempt to find a general chain rule formula for differentiating the composition f ◦ g of Lipschitz functions f and g that would be as close as possible to the standard formula (f ◦ g)'(x) = f'(g(x)) ◦ g'(x), we show that this formula holds without any artificial assumptions provided derivatives are replaced by complete derivative assignments. The idea behind these assignments is that the derivative of f at y is understood as defined only in the direction of a suitable “tangent space” U(f, y) (and so it exists at every point), but these tangent spaces are chosen in such a way that for any g they contain the range of g'(x) for almost every x. Showing the existence of such assignments leads us to a detailed study of derived sets and the ways in which they describe pointwise behavior of Lipschitz functions.