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Rev. Acad. Canar. Cienc., XII (Núms. 1-2), 9-17 (2000) - (Publicado en julio de 2001)
B*-CONTINUITY AND OTHER GENERALISED CONTINUITY
D. K Ganguly and Chandrani Mitra
ABSTRACT The paper is devoted to study sorne properties of generalized types of continuity
and B*-continuity and sorne examples are given in this connection.
INTRODUCTION AND PRELIMINARY EXAMPLES:
Continuous function is a source of studies for many of the research workers and it has been
generalized in various ways, viz. quasi-continuity, cliquishness, simply -continuity, B-continuity,
almost continuity, semi-continuity in the sense of Levine, functions having the WCIVP etc. It is
seen that the notions of almost continuity, quasi-continuity and semi-continuity in the sense of
Levine are same [ 4 ]. These functions involve the concept of generalized open sets, viz. semiopen,
simply open etc. In recent years a good number of research has been done on many types
of generalized continuity. Perhaps the notion of quasi-continuity has been focussed most
intensively due to its close connection with other types of continuity and its various applications
in Topology and Mathematical Analysis. The notion of B*-continuous function has been
introduced by making sorne modification in the definition of quasi-continuous function and it has
a closed interrelation with this function. B*-continuity is more general than quasi-continuity.
Our terminology is standard. Unless otherwise stated X and Y are topological spaces. We denote
the closure and interior of a subset A of X by el (A) and int (A). The following is a Iist of the
definitions that are used in our paper.
1991 AMS Subject Classifications: Primary-54C08, Secondary-26A15.
Keywords & Phrases: Quasi-continuity, Cliquishness, Semi-continuity, Semi-open· set, Simplycontinuity,
Almost continuity, B- continuity, B*-set, B*-continuity, WCIVP ..
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Definition 1:
+ A function f: X --+ Y is quasi-continuous at a point x E X if, for each open neighborhood U
of x and each open neighborhood V off (x), there is a nonempty open set G e U such that
f(G) e V.
Iffis quasi-continuous at each point it is said to be quasi-continuous on X. [ 4]
+ A set A e X is said to be semi-open iff3 an open set O such that O e A e el (O).
+ A function f: X--+ Y is semi- continuous in the sense ofLevine if for any open set V in Y,
r 1(V) is semi-open in X. [ 6]
+ A function f: X --+Y is simply continuous if for each open set V in Y, the set r 1(V) is the
union of an open set and a nowhere dense set in X. [ 2 ]
+ A function f: X --+Y is called almost continuous at a point p E X, if for any neighborhood V
ofthe point f(p) in Y we have p E int (el (r1(V))).
Iffis almost continuous at each point then it is said to be so on the whole ofX. [ 8]
+ A function f: X --+ Y (Y is a metric space with metric d) is cliquish at a point xEX if for each .
e > O and each open neighborhood U ofx, there is a nonempty open set Ge U such that,
d (f(x1), f(x2)) <e V x1, x2 E G.
If f is cliquish at each point, it is said to be so over the whole of X. [ 8 ]
We recall that a set A has the property ofBaire if A can be expressed as A= Gd P, where Gis
open and P is of first category.
+ A function f: X --+ Y is said to be B-continuous at x E X, if for any open neighborhood U of
x and V off (x), there is a set B which is either open or second category having the Baire
property such that B e Un C 1(V).
If f is B-continuous at each point of X it is said to be B-continuous over X. [ 7 ]
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The goal of this paper is to introduce a notion of B * continuity and present sorne results relating
to various kinds of generalized continuous functions.
Definition 2: A set A e X is said to be B* set if it is not nowhere dense in X and have the
property ofBaire.
Definition 3: A function f: X -+ Y is said to be B * continuous at x if for each open set U
containing x in X and each open set V containing f (x) in Y, there is a B* set such that B e Un
r-1(V).
Iffbe B* continuous at each point ofX, then fis said to be B* continuous on the whole ofX.
Definition 4: A function f: [a, b] -+IR is said to have the WCIVP (Weak Cantor Intermediate
Value Property ), if for p and q in [a, b] with p * q and f (p) * f ( q), there exists a Cantor set C
between p and q such that f(C) lies between f(p) and f(q). [ 11]
Definition 5: A perfect road of a function f at a point x is a perfect set P such that,
i) x is a bi-lateral point of accumulation of P, and
ii) ~Pis continuous at x. [ 3 ]
Among continuity, quasi continuity, simply continuity, cliquishness, B-continuity and B*-
continuity we have the fol/owing implication:
Continuity
~ Quasi Continuity
~ ~
Simply Continuity B-continuity
Cliquishness B*-Continuity
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The following discussion shows that there is no other implication among the functions:
Example 1: A function which is B*- continuous but not quasi continuous.
Let the function f: [O, 1]-+ [O, 1] be such that,
{
o ifx is rational
f(x) =
1 ifx is irrational
Let Xo E [O, 1] be irrational point, and O< e< l. Then for any open neighborhood U ofXo and any
open set Ge U, there always exist rational points in G.
Then, 1 f (x) - f (Xo) 1 = 1 - O = 1 > a, where xE G, x rational. Thus f is not quasi continuous at X().
But f is B*- continuous. In fact, Xo being irrational, for any neighborhood V off (Xo), f - 1(V)
contains ali the irrational points in [O, 1], which is dense in [O, 1] and has the property ofBaire.
Similar arguments hold for the rational points.
Example 2: A function which is simply continuous but not B*- continuous.
Let f: IR -+ [O, 1] be such that,
f(x) = {
O ifx= 1,2, ... ,n (nis fixed and finite)
elsewhere
Then, fis clearly simply continuous. But fis not B*- continuous at each of 1,2, .. .,n.
Remark 1: The above function in example 2 is also cliquish.
Remark 2: The function in example 1 is B*- continuous but is neither cliquish nor simply
continuous.
Example 3: Functíon which is B*- continuous but not B- continuous.
We have seen that Dirichlet's function is B*- continuous. Now, we consider any open
neighborhood V of O, not containing 1 in IR. Then, r-1(V) is the set of ali rational points of [O, 1].
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Now, for any rational p in [O, 1] and any neighborhood U of p, ~ any seeond eategory set in U n
C 1(V). Henee, f is not B-continuous.
MAIN RESULTS:
Tbeorem 1: Jf f is B *- continuous then,
i) For each open set V o/Y with f (x) E V, 3 a semi open set O containing x, such that
O e el (f- 1(V));
ii) For each neighborhoodV off(x), x E el (int (el (C1(V)))).
Proof: i) For eaeh open neighborhood U ofx and eaeh open set V in Y, with f(x)EV, int (U) n
r -1(V) is not nowhere dense. Henee we have, (rp *) int (el (f -1(V) n int (U))) e el (int (U)).
Put G = int (el (f -1(V) n int (U))) n int (U). Then G (*'P) is an open set sueh that Ge U and Ge
el (f-1(V)).
Let V be any open set containing f (x). U., be the family of open neighborhood of x. For eaeh U
E Ux. :l an open set GuofX sueh that Gu (*'P) e U and Gu e el (C1(V)).
Set W =U {Gu}. Then W is open set of:X, x E el (W) and W e el (C1(V)). Take O= W U {x} .
Then, W e O e el (W). So, O is a semi open set and O e el (f - 1(V)).
ii) Let N be any neighborhood off (x). Then, there exists semi open set U eontaining x sueh that
U e el (C1(V)). Therefore, x E el (int (U)) ccl (int (el (C1(V)))).
Theorem 2: Let f: X --+ Y, then the fo/lowing conditions are equivalent:
i) For every dense set D, f(el (D)) e el (f(D)),
ii) f is quasi continuous.
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Proof: i) => ii)
Let p E X and U is an open set in X containing p, and V be an open set in Y containing f (p ). Iff
is not quasi continuous at p, then U would not contain any open set G such that f(G) e V.
Let D = { x E U: f(x) ff. V}. Then D is dense in U. Then f(el (D)) e el (f(D)). Now p E el (D) =>
f(p) E f(el (D)) e el (f(D)). =>V n f(D) #p. But, this contradicts the construction ofD. Hence f
must be quasi continuous.
ii) => i) Let f be quasi continuous. If possible, !et f (el (D)) ~ el (f (D)), where D is a dense set.
Let p E el (D). Then f (p) ff. el (f (D)). Then for V (open in Y) 3 f (p ), Vn f (D) = rp.
Now f being quasi continuous, for any open set U 3 p, 3 a nonempty open set G e U such that
f(G) e V. Again D being dense, Dn G #p. Then, Vn f(D) #p.
This contradiction proves that f (el (D)) e el (f (D)).
Remark 3: H Blumberg showed that [l] for every real-valued function defined on IR, there exists
a dense sub set B of IR such that ~Bis continuous. If f: X --+ Y where X & Y are metric spaces and
Bis dense subset ofX such that ~ 8 is continuous, then we say that Bis a Blumberg set off A set
D e X is called strongly Blumberg set for the function f if D is a Blumberg set for f and for
every open set Ge X, the set f(G n D) is dense in f(G). (9]
J. C. Neugebauer established a elose relation between quasi-continuity and Blumberg set (9]. The
result states that: A function f: [O, l] --+ IR possesses a strongly Blumberg set iff f is quasicontinuous.
Applying these facts to our consideration we have the following corollaries of the
theorem.
Corollary 1: A junction f: [O, 1] --+ IR possesses a strongly Blumberg set iff for every dense set
D, f(el (D)) e el (f(D)).
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Corollary 2: A Junction f: X -+ Y, having the Darboux property has a perfect road if jor every
dense set D, f (el (D)) e el (f (D)).
This follows from the result [5] that a quasi-continuous function having the Darboux propeny
has a perfect road.
Theorem 3: If Q (t) is the set of points of quasi continuity of the Junction f: X -+ Y and X\ Q (t)
is nowhere dense, then f is simply continuous.
Proof: Let V be an open set in Y. Then, we know that [ 2 ], Q (t) n (f - 1(V) \ int f - 1(V)) is
nowhere dense. Then the set C 1(V) \ int C 1(V) e Q (t) n (f -1(V) \ int f-1(V)) U (X \ Q (t)) is
nowhere dense. Hence, fis simply continuous.
Theorem 4: A simply continuous Junction f: X-+ Y, is B *- continuous if for each open set V in
Y, C 1(V) contains a nonempty open set.
Proof: Let f be simply continuous at a point Xo E X. Then for each neighborhood V off (Xo) and
for each neighborhood U of Xo the set f -1(V) is the union of a nonempty open set G and a
nowhere dense set. Then Un C 1(V) contains a nonempty open set and a nowhere dense set.
Thus, Un C 1(V) contains a B*-set. Hence f is B*- continuous.
We know that a function is in DBI elass (Baire one functions having the Darboux property) iff it
has a perfect road at each of its points [ 3 ]. W e can construct a B * - continuous functions with a
perfect road at each point such that the function fails to do have the Darboux property.
Example 4: Consider the function f: [O, l]-+ [O, l] such that,
f(x)=
if XEA
if xEB
if xEC
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Where, A is the union of the closure of all the open intervals that are removed in the odd stage in
the construction of the Cantor set,
B is the union of the closure of all the open intervals that are removed in the even stage in the
construction of the Cantor set,
and C is the non end points points of the Cantor set.
Now, A and B are perfect sets. Clearly, fhas a perfect road at each of its points. Again fis quasi
continuous at every point of [O, 1 ], hence, f is B *- continuous over [O, 1].
But f ([O, 1]) = {O, 1}. Hence f is not Darboux.
We show that there are B*- continuous functions with perfect road at each point, but the
functions do not possess the WCIVP.
Example 5: Let f: [O, 7t] -+ [O, l] be such that, f (x) = sin x. Then f is continuous and so B*continuous
over [O, 7t]. Also f has a perfect road at each of its points. Now, consider the points
7t/3 and 37t/4. Then f(7t/3) = --J3¡2 :t: 11--J2 = f(37t/4). We construct a Cantor set C between 7t/3 and
37t/4. But f(C) will not lie between 11--J2 and --J3/2.
1. Blumberg. H.
2. Borsik. J., Dobos. J.
3. Bruckner. A M.
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