Given a collection of functions of some class defined on the real line, when can you find a large set upon which the restriction of every function is continuous? We consider this problem (and related problems) for various classes of functions and various notions of largeness. These problems can be considered in terms of finding the covering, uniformity (non), additivity, and cofinality numbers for some ideal-like collections of sets.
"Generalizing the Blumberg Theorem." Real Anal. Exchange 27 (2) 423 - 440, 2001/2002.