Darboux-Baire-1-function

Darboux–Baire one-function, Darboux function of the first Baire class

A real-valued function of a real variable of the first Baire class (cf. Baire classes) that satisfies the Darboux property.

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 $X \in \mathbf R$ there exist sequences $\{ x _ { n } \}$, $\{ y _ { n } \}$ such that $x _ { n } \nearrow x \swarrow y _ { n }$ and

\begin{equation*} \operatorname { lim } _ { n \rightarrow \infty } f ( x _ { n } ) = f ( n ) = \operatorname { lim } _ { n \rightarrow \infty } f ( y _ { n } ). \end{equation*}

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 $f$ of the first Baire class, the Darboux property of $f$ is equivalent to extendibility of $f$. In 1995, it was proved [a7] that a function $f$ in the first Baire class is a Darboux function if and only if $f$ is first return continuous.

The set of all Darboux functions $f : \mathbf{R} \rightarrow \mathbf{R}$ of the first Baire class will be denoted by $\operatorname{DB} _ { 1 }$. The class $\operatorname{DB} _ { 1 }$ contains many important classes of functions, for example the class $\Delta$ of all (finite) derivatives, the class $\mathcal{A}$ of all Stallings almost-continuous functions, and the class $\mathcal{A} _ { p }$ of all approximately continuous functions (cf. also Approximate continuity). For bounded functions (denoted by the prefix $b$),

\begin{equation*} b \mathcal{A} _ { p } \subset b \Delta . \end{equation*}

One can prove [a5] that in $\text{bDB} _ { 1 }$ (with the metric of the uniform convergence) the sets $b \Delta$ and $b \mathcal{A} _ { p }$ are very small, in fact, they are superporous at each point of $\text{bDB} _ { 1 }$. I. Maximoff proved ([a12], [a13], [a11]) that each function from the larger class ($\operatorname{DB} _ { 1 }$) can be transformed into a function from the smaller class $\mathcal{A} _ { p }$ (or $\Delta$) by a suitable homeomorphic change of variables. In 1961, C. Goffman and D. Waterman considered [a9] connections between $\mathcal{A} _ { p }$ and $\operatorname{DB} _ { 1 }$ for functions mapping a Euclidean space into a metric space.

In 1950, Z. Zahorski considered [a20] the following hierarchy of classes of functions:

\begin{equation*} {\cal M} _ { 0 } = {\cal M} _ { 1 } \supset \ldots \supset {\cal M}_ { 5 } \end{equation*}

Each of these classes is defined in terms of an associated set of a function (the associated sets of $f$ are all sets of the form $\{ x : f ( x ) < \alpha \}$ and $\{ x : f ( x ) > \alpha \}$). The two largest classes $\mathcal{M} _ { 0 }$ and $\mathcal{M} _ { 1 }$ are equal to $\operatorname{DB} _ { 1 }$, the smallest class ($\mathcal{M} _ { 5 }$) is equal to $\mathcal{A} _ { p }$. Zahorski also proved that the class $\Delta$ fits into this "sequence of classes of functions" (if $f \in \Delta$, then $f \in \mathcal{M} _ { 3 }$ and if $f \in b \Delta$; then $f \in \mathcal{M} _ { 4 }$). The similar hierarchy of classes of functions of two variables has been considered in [a14], [a15], [a18].

The class $\operatorname{DB} _ { 1 }$ is closed with respect to uniform convergence. The maximal additive family for $\operatorname{DB} _ { 1 }$ is the class of all continuous functions. The maximal multiplicative family for $\operatorname{DB} _ { 1 }$ is the class of Darboux functions $f$ with the property: If $x _ { 0 }$ is a right-hand (left-hand) discontinuity point of $f$, then $f ( x _ { 0 } ) = 0$ and there is a sequence $\{ x _ { n } \}$ such that $x _ { n } \searrow x _ { 0 }$ (respectively, $x _ { n } \nearrow x _ { 0 }$) and $f ( x _ { n } ) = 0$, [a8]. Of course, $\operatorname{DB} _ { 1 }$ does not form a ring, but for each function $f \in \operatorname{DB} _ { 1 }$ there exists a ring $R \subset \operatorname {DB} _ { 1 }$ containing the class of all continuous functions and $f \in R$ (see, e.g., [a18]).

In 1963, H. Croft constructed [a6] a function $f \in \operatorname{DB} _ { 1 }$ that is zero almost-everywhere but not identically zero. In 1974, a general method for constructing such functions was given ([a1], [a2]): Let $E \subset [ 0,1 ]$ be an $F _ { \sigma }$-set (cf. also Set of type $F _ { \sigma }$ ($G _ { \delta }$)) that is bilaterally c-dense-in-itself. Then there exists a function $f \in \operatorname{DB} _ { 1 }$ such that $f ( x ) = 0$ for $x \in [ 0,1 ] \backslash E$ and $f ( x ) \in ( 0,1 ]$ for all $x \in E$.

Except the standard class $\operatorname{DB} _ { 1 }$, one can also consider the class $\operatorname{DB} _ { 1 } ^ { * }$ ($f \in \operatorname{DB} _ { 1 } ^ { * }$ if $f$ is a Darboux function and for every non-empty closed set $P$ there is an open interval $I$ such that $I \cap P \neq \emptyset$ and $f _ { I \cap P }$ is continuous; see, e.g., [a17]).

References

[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) MR0507448 Zbl 0382.26002
[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 MR0928988 Zbl 0651.26005
[a5]	B. Świątek, "The functions spaces $D B _ { 1 }$ and $A ^ { * }$" Doctoral Diss. Univ. Lódź (1997) Zbl 1069.26005
[a6]	H. Croft, "A note on a Darboux continuous function" J. London Math. Soc. , 38 (1963) pp. 9–10 Zbl 0111.05801
[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 MR1313777 Zbl 0821.26006
[a8]	R. Fleissner, "A note on Baire 1 Darboux functions" Real Anal. Exch. , 3 (1977-78) Zbl 0387.26003
[a9]	C. Goffman, D. Waterman, "Approximately continuous transformations" Proc. Amer. Math. Soc. , 12 (1916) pp. 116–121 MR0120327 Zbl 0096.17103
[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 MR0023895 Zbl 0063.03853
[a12]	I. Maximoff, "Sur la transformation continue de fonctions" Bull. Soc. Phys. Math. Kazan. , 3 : 12 (1940) pp. 9–41 (In Russian) (French summary) MR0015454 Zbl 0063.03850 Zbl 0063.03848
[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) MR0015456 Zbl 0063.03850
[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 $M _ { 2 }$" Časop. Pro Pěst. Mat. , 91 (1966) pp. 389–411 MR0231950 Zbl 0156.06402
[a16]	T. Natkaniec, "Almost continuity" Habilitation Thesis Bydgoszcz (1992) MR1213454 MR1171393 Zbl 0779.54011 Zbl 0760.54007
[a17]	R.J. O'Malley, "Baire$* 1$, Darboux functions" Proc. Amer. Math. Soc. , 60 (1976) pp. 187–192 Zbl 0646.26004
[a18]	R.J. Pawlak, "Darboux transformations" Habilitation Thesis Univ. Lodz (1985) (In Polish) MR0844262 Zbl 0641.26013
[a19]	J. Young, "A theorem in the theory of functions of a real variable" Rend. Circ. Mat. Palermo , 24 (1907) pp. 187–192 Zbl 38.0420.01
[a20]	Z. Zahorski, "Sur la prémiere dérivée" Trans. Amer. Math. Soc. , 69 (1950) pp. 1–54 MR0037338 Zbl 0038.20602