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 approximately 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.

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(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 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.

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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 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 complementary 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 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 / 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

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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 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, ... , «. Obviously

OO OO OO OO „1ÍÍ°A„W= AW forxeU^andforxe/-U^-UUlK«- n=l «=1 B=li=lj 0. There is an index «i such that 2~m+ < r. For i= I,... , m let «(/) denote 1 if x £ I - U£li U/ N,x £ K¿ k n_j+x for i = I, ... , m and i + k > n and |«„(x)|<2-m+1

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Darboux Baire 1 function and hn(An) is finite, so fn = gn+ hn is a Darboux Baire j function. We have also lim / = lim g + lim h„ = g + h = f n—>oo " n—>oo " «—>oo " and the proof is complete. Remark 4. Suppose that f: I ~* R is a Darboux Baire \ function and Bel is a dense set in I. From Theorem 0 it follows that for every x £ I we have lim inf/(f) < f(x) < lim sup/(r). t€B t€B From [3] we derive the following theorem: Theorem 2. Suppose that F: I —>R is a function such that all the sections Fx(t) = F(x, t) (x, t £ I) are of Darboux Baire \ class (and continuous almost everywhere). If all the sections Fy(t) = F(t,y) (t, y £ I) have the Baire property (are measurable), then F has also the Baire property (is measurable). Example 2. There is a nonmeasurable function F.I -* R such that all the sections Fx , Fy are Darboux Baire \ . Let A c_I be a Cantor set of positive measure and let B c I be a nonmeasurable set of full exterior measure such that all the sections Bx = {t £ I ; (x, t) £ B} and By = {t £ I; (t, y) £ B} are empty or contain one or two points [7]. In every complementary interval (an, bn) of the set A we choose a closed interval In c (an , bn). For every « = 1,2,... we define a continuous function fn: In -* R such that f„{I„) = [0, 1] and fn(x) = 0 if x is an endpoint of In . Define for (x,y)£(Ax(R- A)) U ((R - A) x A), ' fn(y) for x 6 A and y £ In , «=1,2,..., g(x,y) = < f„(x) fory£A&náx£ln, «=1,2,..., 0 in the remaining case. Since the set (Ax (R- A)) U ((R - A) x A) is closed in the space H = (I x I) - (A x A), so there exists a continuous function «: H —>[0, 1] such that h(x, y) = g(x, y) for (x, y) £ (A x (R -A)) U ((R - A) x A). Let ' «(x,y) for (x,y) £H, F(x,y) = \l for (x,y) £ (A xA)r\B, 0 in the remaining cases. Since the set (AxA)nB is not measurable, so F is nonmeasurable. Evidently all the sections F , Fy are in the Darboux Baire \ class.

References

1. Z. Grande, Sur la quasi-continuité et la quasi-continuité approximative, Fund. Math. 129 (1988), 167-172. 2. C. Kuratowski, Topologie I, PWN Warszawa, 1958.

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3. E. Marczewski and Cz. Ryll-Nardzewski, Sur la mesurabilité des fonctions de plusieurs vari- ables, Annales Soc. Polon. Math. 25 (1953), 145-154. 4. R. J. O'Malley, Approximately differentiablefunctions. The r-topology, Pacifie J. Math. 72 (1977), 207-222. 5. D. Preiss, Limits of approximately continuous functions, Czechoslovak Math. J. 96 (1971), 371-372. 6. H. Rosen, Darboux Baire - .5 function, Proc. Amer. Math. Soc. 110 (1990), 285-286. 7. W. Sierpiñski, Sur un problème concernant les ensembles mesurables superficiellement, Fund. Math. 1 (1920), 112-115. 8. E. Stroñska, Algebry generowane przez pewne rodziny funkcji quasi ciqglych, Doctoral thesis, Lodz University, 1990.

Institute of Mathematics, Copernicus University, ul. Chopina 12/18, 87-100 Torun, Poland

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