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SOME REMARKS ABOUT ROSEN's FUNCTIONS C-Dense in the Graph 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, ..
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