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

Maleva, Olga; Preiss, David

doi:10.1090/tran/6480

Directional upper derivatives and the chain rule formula for locally Lipschitz functions on Banach spaces
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by Olga Maleva and David Preiss PDF

Trans. Amer. Math. Soc. 368 (2016), 4685-4730 Request permission

Abstract:

Motivated by an attempt to find a general chain rule formula for differentiating the composition $f\circ g$ of Lipschitz functions $f$ and $g$ that would be as close as possible to the standard formula $(f\circ g)’(x) = f’(g(x))\circ g’(x)$, 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.

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