The classical inverse/implicit function theorem revolves around solving an equation involving a differentiable function in terms of a parameter and tells us when the solution mapping associated with this equation is a differentiable function. Already in 1927 Hildebrand and Graves observed that one can put aside differentiability using instead Lipschitz continuity. Subsequently, Graves developed various extensions of this idea, most known of which are the Lyusternik-Graves theorem, where the inverse of a function is a set-valued mapping with certain Lipschitz type properties, and the Bartle-Graves theorem which establishes the existence of a continuous and calm selection of the inverse. In the last several decades more sophisticated results have been obtained by employing various concepts of regularity of mappings acting in metric spaces, mainly aiming at applications to numerical analysis and optimization. This paper presents a unified view to the inverse function theorems that originate from the works of Graves. It has a historical flavor, but not entirely, tracing the development of ideas from a personal perspective rather than surveying the literature.
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The author wishes to thank Radek Cibulka for his help when preparing this paper. This work was supported by the National Science Foundation (NSF) Grant 156229, the Austrian Science Foundation (FWF) Grant P31400-N32, and the Australian Research Council (ARC) Project DP160100854.
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