proceedings of the american mathematical society Volume 113, Number 1, September 1991 SOME REMARKS ABOUT ROSEN'S FUNCTIONS ZBIGNIEW GRANDE (Communicated by Andrew M. Bruckner) Abstract. The main result is: Each Baire 2 function /:/—»/? whose set of continuity points is dense is the pointwise limit of a sequence of Darboux Baire \ functions. Let / = [0, 1] and R be the set of all reals. A function /:/—>/? is said to be Baire \ [6] if preimages of open sets are G^-sets. (Rosen states these functions as Baire -.5 [6].) In [6] H. Rosen shows the following theorem: Theorem 0. Suppose f: I —► R is a Darboux Baire \ function, and let D denote the set of points at which f is continuous. Then the graph of f/D is bilaterally c-dense in the graph of f. Remark 1. A function f:I—>R is a Baire j function iff it is ambiguously a Baire 1 function, i.e. preimages of open sets are Gs- and i^-sets simultaneously. Indeed if / is Baire \ , then every open set U is the sum of closed sets Fn (n - 1, 2, ...), hence (oo \ oo n=l ) n=l and U~, r\Fn) is an F„-set. Remark 2. A function /:/->Ä is said to be a.e. continuous [4] if it is ap- proximately continuous and continuous almost everywhere (in the sense of the Lebesgue measure). There is an a.e. continuous function f:I—>R which is not Baire j. Example 1. Indeed, let P c / be a Cantor set of measure zero and let A c P be a countable set such that CIA = P (CIA denotes the closure of the set A). Let (an)n be a sequence of all points of the set A . There is a family of closed intervals Jnm c I - P (n, m = 1, 2, ...) such that: (!) JnmnJrs = 0 ü (n, m) ¿ (r, s), n, m, r,s = 1, 2, ... ; (2) an is a density point of the set (Jm=i Km > n = 1,2, ... ; Received by the editors March 9, 1990. 1980 Mathematics Subject Classification (1985 Revision). Primary 26A21; Secondary 26A03, 26B05, 28A20. © 1991 American Mathematical Society 0002-9939/91 $1.00+ $.25 per page 117 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use