The following example is interesting in light of the previous theorem.
Example 1.1.5. Let I be the σ-ideal consisting of all first category subsets of R. I-continuity is often called qualitative continuity [26]. It is well-known in this case that f is a Baire function if, and only if, f is qualitatively continuous I-a.e.
In particular, combining Example 1.1.5 with Theorem 1.1.4 yields the following well-known corollary, which will be useful in the sequel.
Corollary 1.1.6. Let f : R → R. The following statements are equivalent. (i): f is a Baire function. (ii): There exists a residual set K such that f|K is continuous.*2 (iii): f is qualitatively continuous I-a.e. In the case of Lebesgue measure, the following is true.
*2 A set is residual if its complement is first category. This is often called comeager.
