まず、関連参考：検索でヒットしたので貼る。 BaireCategory.pdfの”3. Pointwise limits of continuous functions.”に、「422に書いた定理」の関連記述 「Theorem. If f : R → R is a pointwise limit of continuous functions, then Df is Fσ meager (that is, a countable union of closed sets with empty interior). (In particular, by Baire's theorem, f is continuous on a dense subset of R.)」とあり（当たり前か？ （＾＾ ）
3. Pointwise limits of continuous functions. Theorem. If f : R → R is a pointwise limit of continuous functions, then Df is Fσ meager (that is, a countable union of closed sets with empty interior). (In particular, by Baire's theorem, f is continuous on a dense subset of R.)
Proof. We know Df = ∪ n>=1 D1/n (see Section 1), so it suffices to show that the closed sets Dε have empty interior, for any ε > 0. By contradiction, suppose Dε contains an open interval I. We'll find an open interval J ⊂ I disjoint from Dε! Let fn → f pointwise on R, with each fn : R → R continuous. For each N >= 1, consider the set: CN = {x ∈ I; (∀m, n >= N)|fm(x) - fn(x)| <= ε/3}. Clearly ∪ N>=1 CN = I (by pointwise convergence). QED (引用終り)
