  In this article we treat a notion of continuity for a multi-valued function F
and we compute the descriptive set-theoretic complexity of the set of all x for
which F is continuous at x. We give conditions under which the latter set is
either a G_\delta set or the countable union of G_\delta sets. Also we provide
a counterexample which shows that the latter result is optimum under the same
conditions. Moreover we prove that those conditions are necessary in order to
obtain that the set of points of continuity of F is Borel i.e., we show that if
we drop some of the previous conditions then there is a multi-valued function F
whose graph is a Borel set and the set of points of continuity of F is not a
Borel set. Finally we give some analogue results regarding a stronger notion of
continuity for a multi-valued function. This article is motivated by a question
of M. Ziegler in "Real Computation with Least Discrete Advice: A Complexity
Theory of Nonuniform Computability with Applications to Linear Algebra",
(submitted).
