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

The descriptive set-theoretic complexity of the set of points of continuity of a multi-valued function (Extended Abstract)