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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 detailed study of derived sets and the ways in which they describe pointwise behavior of Lipschitz functions.

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