# 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]).

How to Cite This Entry:
Darboux-Baire-1-function. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Darboux-Baire-1-function&oldid=50031
This article was adapted from an original article by R.J. Pawlak (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article