Definition
Partial regularity
Regularity away from an exceptional set whose size is quantitatively controlled.
A partial regularity theorem specifies a class of solutions, a notion of regular point, and a bound on the size of the exceptional set of nonregular points. A typical conclusion is , using Hausdorff measure or, for parabolic equations, parabolic Hausdorff measure. The regular points usually form an open set.
Meaning of the conclusion
Small measure or dimension of the exceptional set does not imply that it is empty. The Caffarelli–Kohn–Nirenberg theorem is a standard example.