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The book is about differentiability of six operators on functions or pairs of functions: composition (f of g), integration (of f dg), multiplication and convolution of two functions, both varying, and the product integral and inverse operators for one function. The operators are differentiable with respect to p-variation norms with optimal remainder bounds. Thus the functions as arguments of the operators can be nonsmooth, possibly discontinuous, but four of the six operators turn out to be analytic (holomorphic) for some p-variation norms. The reader will need to know basic real analysis, including Riemann and Lebesgue integration. The book is intended for analysts, statisticians and probabilists. Analysts and statisticians have each studied the differentiability of some of the operators from different viewpoints, and this volume seeks to unify and expand their results.

Differential operators --- Functions of bounded variation --- Differentiaaloperatoren --- Fonctions à variation bornée --- Functies met begrensde variatie --- Integralen --- Integrals --- Intégrales --- Operateurs differentiels --- Operator theory. --- Global analysis (Mathematics). --- Manifolds (Mathematics). --- Functions of real variables. --- Operator Theory. --- Global Analysis and Analysis on Manifolds. --- Real Functions. --- Real variables --- Functions of complex variables --- Geometry, Differential --- Topology --- Analysis, Global (Mathematics) --- Differential topology --- Geometry, Algebraic --- Functional analysis