Darboux–Baire one-function, Darboux function of the first Baire class
In the first Baire class, the Darboux property is known to be equivalent to other properties. For example, in 1907, J. Young considered [a19] the following property: For each there exist sequences , such that and
He proved that for functions of the first Baire class, the Darboux property and this Young property are equivalent. In 1922, K. Kuratowski and W. Sierpiński proved [a10] that for real-valued functions of the first Baire class and defined on an interval, the Darboux property is equivalent to the fact that the function has a connected graph (cf. also Graph of a mapping). In 1974, J. Brown showed [a3] that for real functions of the first Baire class and defined on an interval, the Darboux property is equivalent to Stallings almost continuity. In 1988, it was shown [a4] that for a function of the first Baire class, the Darboux property of is equivalent to extendibility of . In 1995, it was proved [a7] that a function in the first Baire class is a Darboux function if and only if is first return continuous.
The set of all Darboux functions of the first Baire class will be denoted by . The class contains many important classes of functions, for example the class of all (finite) derivatives, the class of all Stallings almost-continuous functions, and the class of all approximately continuous functions (cf. also Approximate continuity). For bounded functions (denoted by the prefix ),
One can prove [a5] that in (with the metric of the uniform convergence) the sets and are very small, in fact, they are superporous at each point of . I. Maximoff proved ([a12], [a13], [a11]) that each function from the larger class () can be transformed into a function from the smaller class (or ) by a suitable homeomorphic change of variables. In 1961, C. Goffman and D. Waterman considered [a9] connections between and for functions mapping a Euclidean space into a metric space.
In 1950, Z. Zahorski considered [a20] the following hierarchy of classes of functions:
Each of these classes is defined in terms of an associated set of a function (the associated sets of are all sets of the form and ). The two largest classes and are equal to , the smallest class () is equal to . Zahorski also proved that the class fits into this "sequence of classes of functions" (if , then and if ; then ). The similar hierarchy of classes of functions of two variables has been considered in [a14], [a15], [a18].
The class is closed with respect to uniform convergence. The maximal additive family for is the class of all continuous functions. The maximal multiplicative family for is the class of Darboux functions with the property: If is a right-hand (left-hand) discontinuity point of , then and there is a sequence such that (respectively, ) and , [a8]. Of course, does not form a ring, but for each function there exists a ring containing the class of all continuous functions and (see, e.g., [a18]).
In 1963, H. Croft constructed [a6] a function that is zero almost-everywhere but not identically zero. In 1974, a general method for constructing such functions was given ([a1], [a2]): Let be an -set (cf. also Set of type ()) that is bilaterally c-dense-in-itself. Then there exists a function such that for and for all .
Except the standard class , one can also consider the class ( if is a Darboux function and for every non-empty closed set there is an open interval such that and is continuous; see, e.g., [a17]).
|[a1]||S. Agronsky, "Characterizations of certain subclasses of the Baire class 1" Doctoral Diss. Univ. Calif. Santa Barbara (1974)|
|[a2]||A.M. Bruckner, "Differentiation of real functions" , Springer (1978)|
|[a3]||J.B. Brown, "Almost continuous Darboux functions and Reed's pointwise convergence criteria" Fund. Math. , 86 (1974) pp. 1–7|
|[a4]||J.B. Brown, P.L. Humke, "Measurable Darboux functions" Proc. Amer. Math. Soc. , 102 : 3 (1988) pp. 603–610|
|[a5]||B. Świątek, "The functions spaces and " Doctoral Diss. Univ. Lódź (1997)|
|[a6]||H. Croft, "A note on a Derboux continuous function" J. London Math. Soc. , 38 (1963) pp. 9–10|
|[a7]||U.B. Darji, M.J. Evans, R.J. O'Malley, "First return path systems: differentiability, continuity and orderings" Acta Math. Hung. , 66 (1995) pp. 83–103|
|[a8]||R. Fleissner, "A note on Baire 1 Darboux functions" Real Anal. Exch. , 3 (1977-78)|
|[a9]||C. Goffman, D. Waterman, "Approximately continuous transformations" Proc. Amer. Math. Soc. , 12 (1916) pp. 116–121|
|[a10]||K. Kuratowski, W. Sierpiński, "Les fonctions de classe 1 et les ensembles convecs punctiformes" Fund. Math. , 3 (1922) pp. 303–313|
|[a11]||I. Maximoff, "On continuous transformation of some functions into an ordinary derivative" Ann. Scuola Norm. Sup. Pisa , 12 (1943) pp. 147–160|
|[a12]||I. Maximoff, "Sur la transformation continue de fonctions" Bull. Soc. Phys. Math. Kazan. , 3 : 12 (1940) pp. 9–41 (In Russian) (French summary)|
|[a13]||I. Maximoff, "Sur la transformation continue de quelques fonctions en dérivées exactes" Bull. Soc. Phys. Math. Kazan. , 3 : 12 (1940) pp. 57–81 (In Russian) (French summary)|
|[a14]||L. Mišik, "Über die Eigenschaft von Darboux und einiger Klassen von Funktionen" Rev. Roum. Math. Pures Appl. , 11 (1966) pp. 411–430|
|[a15]||L. Mišik, "Über die Klasse " Časop. Pro Pěst. Mat. , 91 (1966) pp. 389–411|
|[a16]||T. Natkaniec, "Almost continuity" Habilitation Thesis Bydgoszcz (1992)|
|[a17]||R.J. O'Malley, "Baire, Darboux functions" Proc. Amer. Math. Soc. , 60 (1976) pp. 187–192|
|[a18]||R.J. Pawlak, "Darboux transformations" Habilitation Thesis Univ. Lodz (1985) (In Polish)|
|[a19]||J. Young, "A theorem in the theory of functions of a real variable" Rend. Circ. Mat. Palermo , 24 (1907) pp. 187–192|
|[a20]||Z. Zahorski, "Sur la prémiere dérivée" Trans. Amer. Math. Soc. , 69 (1950) pp. 1–54|
Darboux-Baire-1-function. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Darboux-Baire-1-function&oldid=18540