  We 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.
