zbMATH — the first resource for mathematics

On the averages of Darboux functions. (English) Zbl 0908.26005
Two problems are considered by the author. (1) Characterize families $$\mathcal F$$ of real functions for which there exists a Darboux function $$\psi$$ such that $$\psi >f$$ for each $$f\in{\mathcal F}$$. (2) Characterize families $$\mathcal F$$ of real functions for which there exists two Darboux functions $$\varphi,\psi$$ such that $$\varphi<f<\psi$$ for each $$f\in{\mathcal F}$$. Note that those problems are equivalent to the following questions. $$(1')$$ Does there exist $$g>0$$ such that $$f+g$$ is Darboux for each $$f\in{\mathcal F}$$? $$(2')$$ Does there exist $$g>0$$ such that both $$f+g$$ and $$-f+g$$ are Darboux for each $$f\in{\mathcal F}$$? The author answers both problems $$(1')$$ and $$(2')$$ for families $$\mathcal F$$ with cardinality less than the cofinality of the continuum. Moreover, he proves that if the size of $$\mathcal F$$ is less than the additivity of the measure and all $$f\in{\mathcal F}$$ are measurable then $$g$$ can be also measurable. The similar result is proved for families of functions with the Baire property. In the second part of the paper questions $$(1')$$ and $$(2')$$ are considered in the class of cliquish (i.e., pointwise discontinuous) functions. In the third part of the paper the author answers those questions for finite families of functions from the first class of Baire. As a corollary he obtains the characterization of averages of Darboux functions in the first class of Baire. In particular he shows that not every $$f\in B_1$$ is an average of two $$DB_1$$ functions. This solves a question of A. M. Bruckner, J. G. Ceder and T. L. Pearson [Rev. Roum. Math. Pures Appl. 19, No. 3, 977-988 (19974; Zbl 0289.26005)]. (See also J. G. Ceder and T. L. Pearson’s survey article [Real Anal. Exch. 9, No. 1, 179-194 (1984; Zbl 0579.26002)]).

MSC:
 26A21 Classification of real functions; Baire classification of sets and functions 26A15 Continuity and related questions (modulus of continuity, semicontinuity, discontinuities, etc.) for real functions in one variable 54C30 Real-valued functions in general topology 54C08 Weak and generalized continuity
Citations:
Zbl 0289.26005; Zbl 0579.26002
Full Text:
References:
  Jack B. Brown, Almost continuous Darboux functions and Reed’s pointwise convergence criteria, Fund. Math. 86 (1974), 1 – 7. · Zbl 0297.26004  Andrew M. Bruckner, Differentiation of real functions, Lecture Notes in Mathematics, vol. 659, Springer, Berlin, 1978. · Zbl 0382.26002  A. M. Bruckner and J. G. Ceder, Darboux continuity, Jber. Deutsch. Math.-Verein. 67 (1964/1965), no. Abt. 1, 93 – 117. · Zbl 0144.30003  A. M. Bruckner, J. G. Ceder, and T. L. Pearson, On Darboux functions, Rev. Roumaine Math. Pures Appl. 19 (1974), 977 – 988. · Zbl 0289.26005  A. M. Bruckner and John L. Leonard, Stationary sets and determining sets for certain classes of Darboux functions, Proc. Amer. Math. Soc. 16 (1965), 935 – 940. · Zbl 0141.05901  Jack G. Ceder, Differentiable roads for real functions, Fund. Math. 65 (1969), 351 – 358. · Zbl 0188.12003  J. G. Ceder and T. L. Pearson, Insertion of open functions, Duke Math. J. 35 (1968), 277 – 288. · Zbl 0174.09302  J. G. Ceder and T. L. Pearson, A survey of Darboux Baire 1 functions, Real Anal. Exchange 9 (1983/84), no. 1, 179 – 194. · Zbl 0579.26002  Krzysztof Ciesielski and Arnold W. Miller, Cardinal invariants concerning functions whose sum is almost continuous, Real Anal. Exchange 20 (1994/95), no. 2, 657 – 672. · Zbl 0830.26004  Zbigniew Grande, On the Darboux property of the sum of cliquish functions, Real Anal. Exchange 17 (1991/92), no. 2, 571 – 576. · Zbl 0762.26001  Aleksander Maliszewski, Sums of bounded Darboux functions, Real Anal. Exchange 20 (1994/95), no. 2, 673 – 680. · Zbl 0830.26002  Aleksander Maliszewski, On theorems of Pu & Pu and Grande, Math. Bohem. 121 (1996), no. 1, 83 – 87. · Zbl 0863.26005  I. Maximoff, Sur les fonctions ayant la propriété de Darboux, Prace Mat.-Fiz. 43 (1936), 241-265. · Zbl 0013.25007  -, Sur la transformation continue de fonctions, Bull. Soc. Phys. Math. Kazan 12 (1940), 9-41. · Zbl 0063.03848  -, On functions of class $$1$$ having the property of Darboux, Amer. J. Math. 65 (1943), 161-170. · Zbl 0063.03852  Richard J. O’Malley, Baire* 1, Darboux functions, Proc. Amer. Math. Soc. 60 (1976), 187 – 192. · Zbl 0339.26010  David Preiss, Maximoff’s theorem, Real Anal. Exchange 5 (1979/80), no. 1, 92 – 104. · Zbl 0442.26004  Jaroslav Smítal, On approximation of Baire functions by Darboux functions, Czechoslovak Math. J. 21(96) (1971), 418 – 423. · Zbl 0219.26005  J. Stallings, Fixed point theorems for connectivity maps, Fund. Math. 47 (1959), 249 – 263. · Zbl 0114.39102  J. Young, A theorem in the theory of functions of a real variable, Rend. Circ. Mat. Palermo 24 (1907), 187-192. · JFM 38.0420.01  Z. Zahorski, Sur la première dérivée, Trans. Amer. Math. Soc. 69 (1950), 1-54. · Zbl 0038.20602
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.