Given a Polish space X let J(X) denote the collection of nonempty compact subsets of X with the Hausdorff metric. We investigate the connection between the Borel complexity a function f and the Borel complexity of the set T(f)={C in J(X): f|C is continuous}. Generally, the set T(f) is an ideal. One can see the subject of this paper from at least two directions. First, one can see the complexity of T(f) as a measure of how discontinuous f is, since for f continuous T(f)=J(X) which is a very simple set. Secondly, descriptive set theorist have an interest in finding natural examples of objects such as ideals of compact sets which are complex.

**Last modified June 24, 2002.**