Messerschmidt, Miek2019-05-282019-03Messerschmidt, M. A Pointwise Lipschitz Selection Theorem. Set-Valued and Variational Analysis (2019) 27: 223-240. https://doi.org/10.1007/s11228-017-0455-2.1877-0533 (print)1877-0541 (online)10.1007/s11228-017-0455-2http://hdl.handle.net/2263/69220We prove that any correspondence (multi-function) mapping a metric space into a Banach space that satisfies a certain pointwise Lipschitz condition, always has a continuous selection that is pointwise Lipschitz on a dense set of its domain. We apply our selection theorem to demonstrate a slight improvement to a well-known version of the classical Bartle-Graves Theorem: Any continuous linear surjection between infinite dimensional Banach spaces has a positively homogeneous continuous right inverse that is pointwise Lipschitz on a dense meager set of its domain. An example devised by Aharoni and Lindenstrauss shows that our pointwise Lipschitz selection theorem is in some sense optimal: It is impossible to improve our pointwise Lipschitz selection theorem to one that yields a selection that is pointwise Lipschitz on the whole of its domain in general.en© Springer Science+Business Media B.V. 2017. The original publication is available at https://link.springer.com/journal/11228.Selection theoremPointwise Lipschitz mapBartle-Graves theoremPostprint Article