Hindawi Publishing Corporation
International Journal of Mathematics and Mathematical Sciences
Volume 2013, Article ID 878253, 3 pages
http://dx.doi.org/10.1155/2013/878253
Research Article
Left Baire-1 Compositors and Continuous Functions
Jonald P. Fenecios
1
and Emmanuel A. Cabral
2
1
Department of Mathematics, Ateneo de Davao University, E. Jacinto Street, 8000 Davao City, Philippines
2
Department of Mathematics, Ateneo de Manila University, Loyola Heights, 1108 Quezon City, Philippines
Correspondence should be addressed to Jonald P. Fenecios; jpfenecios@addu.edu.ph
Received 22 February 2013; Revised 23 September 2013; Accepted 26 September 2013
Academic Editor: Hernando Quevedo
Copyright © 2013 J. P. Fenecios and E. A. Cabral. Tis is an open access article distributed under the Creative Commons
Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is
properly cited.
We showed that the class of lef Baire-1 compositors is precisely the class of continuous functions. Tis answers a characterization
problem posed in the work by Zhao (2007, page 550) and settles in the afrmative the conjecture in the work by Fenecios and Cabral
(2012, page 43). Moreover, based on the above result we provide a new proof that the class of functions B
S,T
where S is the class of
all positive continuous functions and T is the class of all positive constants as defned in the work by Bakowska and Pawlak (2010)
is exactly the class of all continuous functions.
1. Introduction
Let (,
) be a complete separable metric space. In the
classical sense a function :→ R where R is the real
line is Baire-1 if, for every open set ⊂ R,
−1
() is an
set in . Equivalently, a function :→ R is Baire-1 if,
for every closed set in , the restriction |
has a point of
continuity in . In fact, the set of points of continuity of is
a residual subset of .
Recently, Lee et al. [1] discovered a new equivalent
defnition of Baire-1 functions. A function :→ R is
Baire-1 if and only if for each >0 there is a positive function
:→ R
+
such that for any ,∈
(,)< min {(), ()} ⇒
()− ()
< .
(1)
Various investigations have been done on the class of Baire-1
functions as well as on its subclasses using the - character-
ization. See, for instance, [2–4].
Let us recall in [4] that the notion of the lef Baire-1
compositors was introduced as a natural counterpart of the
notion of right Baire-1 compositors. A function : R → R is
a lef Baire-1 compositor if for every Baire-1 function :→
R the composition of functions ∘:→ R is Baire-
1. Unlike the right Baire-1 compositors there is no known
characterization for the lef Baire-1 functions. It was shown in
[3] that there exists a function with fnite set of discontinuity
which is not a lef Baire-1 compositor. Tis observation leads
to the conjecture that lef Baire-1 functions must have very
nice properties. We proved that this is indeed the case: the
class of the lef Baire-1 compositors is precisely the class of
all continuous functions defned on R. Moreover, we use this
fact to reestablish that B
C
+
,T
⊂ C where C
+
is the class of
all positive continuous functions, T is the class of all positive
real constants, and C is the class of all continuous functions
on R as defned in [2].
2. Main Result
Let us denote the min{,} by ∧. For easy reference, we
present the following useful results.
In [4, Proposition 1], it was shown that : R → R is
Baire-1 if and only if for every positive continuous function
there is a corresponding positive function such that for any
,∈ R
−
<()∧ ()
⇒
()− ()
<( ())∧(()).
(2)