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 118 ZBIGNIEW GRANDE (3) Cl(U^1/BJ = U^1^U{aJ,« = l,2,...; (4) if x £ I - P, then there is an open neighborhood U of x such that the set of pairs (n, m) with Jnm n U ^ 0 is empty or contains only one element. There are also closed intervals Inm c Int Jnm (Int Jnm denotes the interior of the set Jnm), n, m - 1, 2, ... , such that an is a density point of the set Um=i 7«m' » = 1,2,.... Define, for « = 1,2,..., 2~" for x - an or x 6 7nm, m = 1, 2, ... 0 forxeZ-U^.Int^-K} /»(*) = linear in the component intervals of the set Jnm - lntlnm (m = l,2,...). Obviously every function fn (« = 1, 2, ... ) is a.e. continuous everywhere and continuous at each point x ^ an . So the function oo /-.£/■ n=l is a.e. continuous everywhere and continuous at each point x jí an, « = 1,2,.... Because the set of all points an , « = 1, 2, ... , is not a G5-set, the set {x ; /(x) > 0} is not a G^-set and / is not Baire ¿ . Remark 3. In [5] Preiss showed that every Baire 2 function f:I—*R is the limit of a sequence of approximately continuous functions. From Theorem 0 it follows that every Baire \ function is quasicontinuous, i.e. for every x £ I, for every r > 0 and for every open neighborhood U of x there is a nonempty open set V c U such that \f(u)-f(x)\ <r for all u £ V. (See [1].) So if /: I —>R is the limit of a sequence of Baire j functions then / is a Baire 2 function having dense the set of its continuity points [1]. Our main result is the following Theorem 1. Each Baire 2 function /:/—»/< whose set of continuity points is dense is the pointwise limit of a sequence of Darboux Baire \ functions. Proof. Denote by C(f) the set of all continuity points of /. The set C(f) is a dense C7s-set.There is a Baire 1 function g: I —►R such that g(x) = f(x) for every point x e C(f) [2, p. 342]. Let h = f - g . The set C(h) of continuity points of « is dense and the level set «_1(0) is also dense. Since « is a Baire 2 function, there is a sequence of Baire 1 functions kn: I —>R such that every set kn(I) (« = 1,2,...) is finite and « = lim^^^ [2, pp. 294-295]. For « = 1,2,... denote by An the set of all points x £ R at which osc«(x) > 4~r and put ,(x) = f0 forx£l-An, "[ ! \kn(x) forx£An. License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use REMARKSABOUT ROSEN'S FUNCTIONS 119 We may assume without loss of generality that An+X- An ^ 0 (« = 1,2,...) and |rc„(x)| < 2~" for x e I - An . Evidently « = lim^^^/^ and /„ is continuous at every x £ I - An («=1,2,...). There is a family of closed intervals Kxk! (k — 1,2, ... and I < k) such that: (1) KxklcI-Ax+k+l,k= 1,2,... and I <k; (2) Kmr\KXrs = 0 for (k, l) ¿ (r, s), k, r= 1, 2, ... and /< k, s <r; (3) for every x £ Ax and for every closed interval U 9 x and for every / there is k > I such that Kxk¡ c Int U ; (4) for every x £ I - Ax there is an open neighborhood V of x such that the set of pairs (k, I) with Kxkl (~)V^ 0 is empty or contains only one element. For the construction of such a family it suffices to choose in every comple- mentary interval (an ,bn) of the set Ax two sequences of closed intervals [cink> dlnk] and [c2nk, d2nk\ such that: (!) C2nl > dui > Clnl > d\n2 > Cln2 >-► ä„ ; (2) c2nX < d2nX < c2n2 <d2n2<->bn; (3) lcink> dink\ n Af* =0, k=l,2,... and i =1,2. (See also [8, Lemma 2.1].) For every k = 1,2, ... we define a continuous function lxk: Kxkx—► [-k, k] such that l]k(K[kl) = [~k, k] and llk(x) = 0 if x is an endpoint of the interval Kxkx . Let hAx)= , lxk(x) if x £Kxkx,k= 1,2, ... ( /, (x) in the remaining case. Evidently hx is a center Darboux Baire \ function, because it is continuous at every point x £ I - Ax and hx/Ax is a Baire 1 function such that the set hx(Ax) is finite. In the second step we choose a family K2kl (k = 1,2,... and I < k) of closed intervals such that: (1) K2k! c I - A2+k+¡, k=l,2,... and l<k; (2) K2kl n K2rs = 0 if (k, /) ¿ (r, s), k,r = 1,2,... and I < k and s < r; (3) K2kln Ä"j„ = 0, fc, r = 1, 2, ... and / < k and s < r ; (4) for every x e Cl(^2 - Ax) and for every closed interval V 9 x and for every / there is /c > / such that K2kl c Int F ; (5) for every x £ I - A2 there is an open set U B x such that the set of couples (k, I) with K2kl n U ^ 0 is empty or it contains only one element. For every k = 2,3, ..., we define continuous functions /21i:: A^lyt2—► [-A:, Ä:] and /22A::K2kx -* [-2-1, 2_1] such that l2]k(Kxk2) = [-k, k], l22k(K2kx) = [-2-1, 2~ ], l2lk(x) = 0 for x an endpoint of Kxk2 and l22k(x) = 0 for x License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 120 ZBIGNIEW GRANDE an endpoint of K2kx. Let (l2xk(x) forx € Kxk2, k = 2, 3, ... l22k(x) forx £K2kx, k = 2, 3, ... /2(x) in the remaining case. The function h2 is continuous at each point x £ A2 and it is a Darboux Baire \ function. Generally, in step « > 2 we remark that Kxkl n An = 0 for i < n, k = 1,2,..., / < k with k + I > n and we construct a family of closed intervals Knkl ,k = 1,2, ... and / < k such that: (1) Knk!Cl-An+k+¡,k = l,2,... and /<*; (2) tfnfc/n Äf|rj = 0 if i < « and i + r +s > n ', (k, r = 1, 2, ... , /<&, 5<r); (3) KnklnKnrs = 0 if (k,l)¿(r,s), k,r=l,2,..., l<k, s<r; (4) for every x e Cl(An - An_x), for every closed interval V 9 x and for every / there exists k > I such that Knkl clntV ; (5) for every x £ I - An there is an open set U 9 x such that the set {(&, /) ; Knkl n Í7 ^ 0} is empty or it contains only one element. For every pair (i, A:) such that i + k > n and i < « we define a continuous function lnik: K¡ k n_i+x -* R suchthat lnik(x) = 0 for x being an endpoint of Kink,llnk(Klkn) = [-k,k] and U<*/.*.-w) = t-2_,+1. 2_,+1J for r> ! • Let UW forxei:/.*,«-i+i' z'^"> i + k>n h(x) = { '*(J ln(x) in the remaining case. hn is a Darboux Baire j function, because it is continuous at every point x £ I - An and hJAn is a Baire 1 function such that the set hn(An) is finite and hn(Ak - Ak_x) c (-2"k+\ 2~k+l) for k = 2, 3, ..