A N N ALES SOCIETATIS M A TH EM A TICAE P O LO N A E Series I: C O M M E N T A T IO N E S M ATH EM A TICAE XXVII (1988) ROCZNIKI POLSKIEGO TO W A RZYSTW A M A T E M A T Y C ZN E G O
Séria I: PRACE M A T E M A T Y C ZN E XXVII (1988)
Ryszard Jerzy Pawlak (Lôdz)
On local characterization of Darboux functions
In the paper published in 1965 ([2 ]), A. M. Bruckner and J. G. Ceder introduced the notion of Darboux point of a function / : R - > R . Basic properties o f Darboux points were studied, for example, in papers [3], [8], [11], [12], [13], [15], [21] and [22].
In many papers the notion of Darboux function has been generalized to transformations whose domain and range are topological spaces, more general than the real line (see e.g. [1], [6], [7], [13], [17] and [18]). In this paper we are studying the following notion o f Darboux transformation:
We say that / : X -> Y, where X and Y are arbitrary topological spaces, is a Darboux transformation if f ( L ) is a connected set for every arc i c i ,
It is interesting to remark that if we assume the above generalization of the notion of Darboux function then many theorems connected with notion of Darboux function in the real line are also true in the case our generalization in more abstract spaces than the real line. In particular, we may study ([18]) problems connected with Zahorski classes and approximate continuity for functions of two variables. On the other hand, we may consider new problems which cannot be considered in the case o f real functions o f a real variable. For example, one can prove that in the space of all bounded Darboux functions / : I 2 -> R2 of bounded variation, the set of all discon
tinuous functions is dense and has cardinality T (remark that if / : / -> R is a discontinuous Darboux function, then the variation of / is equal to x>).
Therefore we can investigate the properties o f Darboux transformation / : l 2 ^ bounded variation. We may also consider the problems concern
ing the extension of Darboux functions as well as the theory of homotopy.
In this paper we study generalization of the notion o f a Darboux point in such a way that one can prove a theorem which can be interpreted as a local characterization o f Darboux transformations in arbitrary topological spaces. It is interesting to remark that we can introduce both the definitions of a Darboux point, which for functions / : R - > R are equivalent to the definition in [2 ] as well as the notions which are not equivalent. One can give some applications o f this notion.
We shall use the standard notions and notation. By R we denote the set o f real numbers with the natural topology and by I we denote the unit interval. For an arbitrary function f by C f we denote the set of all
00
continuity points o f /. The symbol V /„(/V ^f) denotes the combination of n= 0
transformations /„, for n = 0, 1, 2, ... (o f / and g): of course, we assume that the considered transformations are compatible. For a function / the image and the inverse image of the intervals (a, b) and [a, b] are denoted by f ( a , b ) , f [ a , b~], f ~ 1(a,b) and / _1[a, b] respectively, to avoid double brackets. Let / be a real function defined on the real line R. By the right (resp. left) side range o f / at x, denoted by R + ( f , x ) (resp. R~ ( f , x)) we mean the set of numbers y such that for every <5 > 0 there exists a point z such that x ^ z < x + <5 (resp. x — Ô < z ^ x) and / (z) = y.
Let X be an arbitrary topological space and let A , B < = X . Then by In t^ B ), ¥ta(B) we shall denote the interior and boundary in the subspace A of X. If A — X then we write In tB and Fr B, respectively. The closure of a set A we denote by Æ
If {x ff} ffeI is a net, then acpxff denotes the set o f all accumulation points
ae Z o f M < x e l -
A subset L e X, where X is an arbitrary topological space, we shall call an arc if there exists some homeomorphism h: I 2ûi^* L. The elements h(0) and h( 1) we shall call the end-points o f /. The arc with the end-points x and y we denote by L (x , y). If L is an arc and a, b e t , then the symbol L L(a, b) denotes an arc with the end-points at a and h, which is contained in L.
Let X be an arbitrary topological space. We say that a nonempty closed set К cuts X (into sets U and V between nonempty sets A and В) if X \ K
= U u V, where U and V are nonempty open sets such that U n V = 0 (and A cz U, В а V). W e say that nonempty set К quasi-cuts a set M с X into sets U and V, between nonempty sets A and B, if M \ K = U u V, where U, V are nonempty separated sets such that A a U and В a V.
In the next definitions we shall denote by / a transformation mapping topological space X into topological space Y.
De f i n i t i o n 1. W e say that a point x 0e X is a Darboux point of the first kind (o f / ) if for every arc L = L( x a, a) the following conditions are fulfilled:
(i) if f ( L L(x0, p)) = Y for every element p e L \ { x 0], then there exists a point Po^t-\{xo] such that / (L , (x 0, p0)) is a connected set;
(ii) if К is a set such that for some net { xffj <= L for which x0 elimx^, К quasi-cuts f ( L ) и a c p / (x j between the sets l/ (x 0)| and
<76 Z <76 Z
I / (x ff): f fe Z ] u a c p / (x (T), then К n / ( L L(x 0, x j ) Ф 0 for every treZ ;
<76 Z
(iii) if for some net {x ff} ffeI c: L for which x0e lim x ff, Y \ f (L) quasi-cuts
<T€l
Characterization of Darboux functions 285
f ( L ) into sets A and В between the sets J/(x0)] and \f(x„): g eZ] in such a way that  n B Ф 0 , then  n B is o f type Gô in the subspace А и В o f Y.
Definition 2. We say that a point x0eX is a Darboux point of the second kind (o f / ) if for every arc / = L (x 0, a) conditions (i) and (ii) of Definition 1 are fulfilled.
Definition 3. We say that a point x0eX is a Darboux point of the third kind (of f ) if for every arc L = L (x 0, a) the following condition if fulfilled:
If К is a set such that for some net [xff} eeî c: L for which x0€ lim xff,
ere
К cuts Y between î/ (x 0)) and { / ( x j : о e Z } u a c p / (x ff), then
(76 2
К r\ f ( Ll (x0, xa)) ф 0 for every o eX.
It is easy to see that if / is some transformation then: if x0 is a Darboux point o f the first (second) kind o f f then x0 is a Darboux point of the second (third) kind o f f
Proposition 4. Let f : X -> Y, where X and Y are arbitrary topological spaces. I f x 0e C f , then x0is a Darboux point of the first kind of f (of course, then x0 is a Darboux point of the second and of the third kind of /).
P r o o f. It suffices to show that condition (i) o f Definition 1 is satisfied.
Suppose, to the contrary, that there exists an arc L = L (x 0, a) such that f ( L t(x0, p ) ) = Y and f ( L t (x0, p ) ) is a disconnected set, for each p e L \ Jx0|.
Let p 0 eL \ Jx, a ) and let L = L t(x0, p 0) . Thus f ( L ) = P kj Q , where P, Q are nonempty separated sets. Assume, for instance, that f ( x 0) e P and oceQ.
Denote by B ( x 0) the local base o f X at x0 consisting of open sets U such that, U n L c z L and B (a) — an arbitrary open local base of Y at a. Write: X
= \(U, V): U £B( x0) and VeB(ct)} and define a relation < by the formula:
( U , V ) ^ ( U ' , V ' ) if and only if U э U' and V u V. Then the relation < directs I .
Now, we define the net {x ff} <T6i in the following way: for every о
= (U, V ) e X let xff denote an arbitrary element of the intersection U n L such that f ( x a)eV. It is easy to see that Xoelim x^ and a e lim / (x (T).
(76 2 (76 2
Since P and Q are separated sets, f ( x 0) e P , <xeQ and a e lim / (x <T), then
(76 2
f (x0) $ \ imf ( x ff) , which contradicts our assumption that x 0 E C f .
(76 2
The next theorem shows that in the case /: R - * R the above definitions are equivalent to the usual definition of a Darboux point (we accept the definition of a Darboux point which is contained in [15]). First we prove the following lemma.
Lemma 5. Let f : R —>R. A point x0eR is a Darboux point o f f of the
th ird kind i f and o n ly i f f o r every a rc L = L ( x , a ), c o n d itio n (ii) o f D e fin itio n 1 is satisfied.
P r o o f. Sufficiency is obvious.
N e ce s s ity . Let L — [ x 0, a] be an arbitrary arc with end-point at xa (assume, for instance, that a > x 0). Let К be a set such that for some net { x a} a e ï c z L for which Xoelimx,,, К quasi-cuts f ( L ) u a cp / (x ff) between the
с т е Х < 7 e X
sets {/ ( x 0)} and { f ( x j : а е Г } u a c p / ( x j . Write:
< 7 e X
(p= sup {y < / ( x 0): y e { f ( x ff) : o e l } u acp/(xff)},
<reX
ф = i n f { y > / ( x 0): y e { f ( x a) : a e l } u a c p / (x j}.
< 7 e X
It is easy to see that (p < f ( x 0) < ф . Now, we shall show that:
(1) if (p > - o o and [ф ,/ (х 0)) c / ( L ) u a c p / ( x <T), ael
then К n [ ( p , f ( x 0) ) Ф 0 . In fact, let ( / (L) u acp f ( x J ) \ K = U u V, where U , V are separated sets
<76 X
such that f ( x 0) e U and {f { x a): a e l j u a c p / (x ff) с V. Suppose, on the
<76 X
contrary, that К n [<р,/(х0)) = 0 . Observe that (peV.. Write q>x = sup \y > (p: [(p, у) a V \ . O f course, [<p, с= V and q> <(p\ < f (x 0). Since
<р х ф У и К, it follows that <P\^U and consequently (p1e V n U , which contradicts Vr\U = 0 . The obtained contradiction ends the proof o f (1).
In a similar way we can prove:
If ф < + oo and (/(x0), ф ] c / ( L ) u a c p / ( x ff), then К r \ ( f ( x 0), ф] Ф 0 .
<76 X
If (P > — o o , then we denote by q>*:
An arbitrary element o f K o ï[<p, / (x0)), if this intersection is nonempty;
An arbitrary element of [<p,f{x0) ) \ ( f ( l.)u a c p / (x ff)), if K n [ ( p , / ( x 0))
<76 X
= 0■
If ф < + o o , then we denote by ф * :
An arbitrary element of К n ( / ( x 0), ф], if this intersection is nonempty;
An arbitrary element o f (/ (x 0), ф] u a c p f { x a)), if К n ( / ( x 0) , > ]
' <76 X
= 0.
Characterization of Darboux functions 287
Moreover, we write
Cep* if (p*eK,
к = \ф* if ф* e K and q> — — oo or (p* фК,
lan arbitrary element o f К in the remaining cases, and
C\tp*, ф*, k] if cp > — oo and ф < + oo, K* = \ k) if (p > — oo and ф = + oo, ' \ф*, k) if cp = — oo and ф < + o o .
One can easily verify that X * cuts Æ between î/ (x 0)î and
\f (x a): a e l ) u a cp f ( x a). Since x0 is a Darboux point o f the third kind, we
<теГ
have K* n f [ x 0, хст] Ф 0 for each g e l . According to definitions of tp*, ф* and K * we infer that К nf\_x0, xf\ Ф 0 , which ends the proof of the lemma.
Th e o r e m 6. Let f : R - + R and let x 0e R . Then the following conditions are equivalent:
(i) x 0 is a Darboux point of the first kind of f (ii) x0 is a Darboux point of the second kind of fi (iii) x 0 is a Darboux point of the third kind of fi (iv) x 0 is a Darboux point of f (in the usual sense).
P r o o f. It is easy to see that (i) =>(ii) =>(iii).
Now, we shall prove the implication (iii) => (iv). We shall show that x0 is a right-sided Darboux point o f / (the proof o f the fact that x0 is a left-sided Darboux point of / is similar).
Let a be an arbitrary real number belonging to (/ (x 0), /?), where /? is some right cluster value o f / at x0 (o f course, if every right cluster value o f / at x0 is equal to / (x 0), then x0 is a Darboux point o f /). Suppose, for instance, that p > f ( x 0). Let {x „} с= / - 1 (/1 — 4(/? — a), f + j ( f — ct)) be a se
quence such that x„ \ x 0 and lim f ( x „ ) = fi. Put K = {ot). Then К
n 00
cuts R between J/(x0)| and j/(x„): n = 1, 2, ...) u acp / (x „). Therefore
n= 1,2,...
K n / [ x 0, x„] # 0 for n —1, 2, . . . , and so oteR + (/, x 0) and consequently x0 is the right-sided Darboux point o f /.
Now, we shov^ that (iv)=>(i). Pass to the proof o f condition (ii) of Definition 1. By virtue o f Lemma 5 it suffices to show that the condition of Definition 3 is satisfied.
Take an arbitrary arc L — [x 0, a] (suppose, for instance, that a > x0 and x0 is the right-sided discontinuity point o f /). Let К be a set such that for some net <= [ * 0, ^or which x0 = limx^, К cuts R between |/(x0)|
and { / (xCT): a e l } и аср/ ( x j . If { / ( x j : a ^ <r0} is a bounded set, for some
<reZ
g0e l , then acp/(xff) # 0 and let f be an arbitrary element o f acp/ (xff). In
ael ereZ
the opposite case, let + oo
P = — x
if for every M > 0 there exists a ^ cr0 such that / ( vJ > M,
if for everyM < 0 there exists <т ^ <r0 such that / ( O < M and there exists M 0 such that f ( x j < M 0 for a ^ (T0.
Suppose, for instance, that fi > f ( x 0). O f course /? is the right cluster value of / at x 0.
According to our assumption, R \ K = U u К where U, V are open sets such that { / ( x 0)} c: U, {/ ( x ff): <t gI } u a cp f ( xa) c= V and U n F = 0 . Put
(TSÎ
a = sup [y > / (x 0): [/ (.v 0), y) c= U J < + o o .
It is easy to see that [/ (x 0), a) <= U and oc^UuV, which means that a gK and f ( x o) < 0L < p . Thus a e R + ( f , x 0) and consequently К n f i x o, Xff] Ф 0 for each a e l .
T o complete the proof it remains to show that condition (i) is satisfied.
Suppose, on the contrary, that there exists an element a such that for every p e ( x 0, d], f [ x 0, p] = R and / [ x 0, p] Ф R. Take a point p0 e(x0, d] (sup
pose, for instance, that a > x 0). Let z e R\ / [x0, p0~\ (suppose that f ( x 0) < z ) . Let n0 be a positive integer such that x 0+\/n0 < p 0. Let
x(I f[.y0, x0+1//?1 n ./- 1 ( - + ^ + x ) for every n ^ n 0. O f course, acp / ( x j = 0 . Thus \z) cuts R between !/ (x 0)| and \f(x„): n = 1, 2, u
n = 1 ,2 ,...
u acp f(x„). Since the condition o f Definition 3 is satisfied, then
n = 1 ,2 ....
!-! n / [x o » Pol Ф 0 , which contradicts our assumption that z ^ / [ x 0, p0].
The obtained contradiction ends the proof o f Theorem 6.
A. M. Bruckner and J. G. Ceder in [2 ] have shown that a real function / is Darboux on I if and only if for every x eI, x is a Darboux point of / (see also [4 ] and [8]). N ow we shall show that a similar theorem is also true for transformations in topological spaces.
Before we prove the first theorem, which is connected with the “ local characterization” o f Darboux transformations, we present the following lemma:
Lemma 7. Let X be an arbitrary topological space and let A, В be separated sets in X. I f Ân В is of type Gô in the subspace A u В of X, then
Characterization o f Darboux functions 289 there exist decreasing sequences \An|, 'Bn] of open sets (in X) such that  <r A„, В cr B„ (for each n) and
Г)Апп П В „ с = X \ ( A v B ) .
n n
P r o o f. Put С = АслВ. Thus C = f]G„, where G„ are open sets in A u В Write G„ = H „ n ( A v B) (for each n), where Hn are open sets in X. Then Г Я „ о ( / 1 и В ) = 0 . Let Щ = H u Щ = H { n H 2, ... O f course
П
X
'H* ) is decreasing sequence of open sets such that С а Г H* and
00 _ _ n= i
n H* n ( A u B) = 0 . Write An = Я * и (Х \ В ), Bn = H * u ( X \ A ) for each n.
n= 1
It is easy to see that {Л „ }, {Bn} are the required sequences.
Th e o r e m 8. Let f : X —► Y, where X and Y are arbitrary topological spaces. Then f is a Darboux transformation if and only if every point x e X is a Darboux point of the first kind of f
P r o o f. Necessity. It is suffices to show that condition (ii) of Definition 1 is satisfied. Let L = L ( x 0,a) be an arbitrary arc in X with the end-point at x0 and let К be an arbitrary set such that for some net î.\„îffe. v c= L for which x0e lim x a, К quasi-cuts f (L) u acp/ ( x j between \f (x0)) and
<xe 2' <7 g 2
\f (x„)\ a eX\ u a c p / (x j. Let cr0 be an arbitrary element o f X. Denote
a g 2
by a* an arbitrary element o f X such that o* ^ o0 and xae L L(x0, xaQ), for each a ^ a*. Since / is a Darboux transformation, it follows that f (L / (x 0, x„0)) is a connected set and К quasi-cuts f (L, (x0, xaQ)) и u acp f ( x j between !/(.y0)| and \f (xa): о ^ cr*j u acp f ( x a), which means that К n f ( L , (x0, x„0)) Ф 0 .
Sufficiency. Suppose, on the contrary, that / is not a Darboux trans
formation. Then there exists an arc / = I (u, v) с: X such that /(/ ) = A u B, where A, В are nonempty, separated sets. Write A’ = f ~ 1(A) n L, B' n L. O f course, A , В are not separated sets. Assume, for instance, that А' слВ' Ф 0 and let д е А ' п В ' . Obviously, дфСf . If q — и (q = v), then we put a = v (a = u). In the case where и Ф q Ф v, we consider two sets M = Lt (q, u) n B ' and N = L t(q, v) n B '. O f course, q e M or q e N and write
a = и
v
if q e M , if q f M .
Let c: B' n L t(q, a) be a net such that gelim x^. If f ( L t(q, p)) = Y for
<re2
each pe L, (q, a) , then there exists an element p0e b L(q,a) such that
f ( L L( q , p 0)) is a connected set. Thus, according to the inequality f ( L t{q, Po)) n A Ф 0 Ф f ( L t{q, p0) ) n B and the inclusion f ( L t {q, p0)) c AkjB, we infer that A, В are not separated, which is impossible. The obtained contradiction proves that there exists an element px e b L{q, a) such that Y \ f ( L L(q, px)) Ф 0 . Write L' = L t(q, px) and let w e Y \ f ( L ' ) . Put A* = A n f ( L ' ) , B* = B n f (L') and let a* be an element of I such that xae t ' , for each Now, we consider two cases:
1. acp f ( x a)r\A* = 0 . Thus {w } quasi-cuts / (£ ') и acp f ( xa) into the
<x>(T* a^a*
sets A* and В* и acp / (xff) between { / (g)} and { / (xff): a ^ <r*} u acp / (xff)
a^a* o^o*
and {w } n /(L ! ) = 0 , which is impossible because q is a Darboux point of the first kind o f f
2. acp / ( x j n Â* Ф 0 . Observe that / (L!) = А* и В* , where f ( q ) e A * , \ f {xa): а ^ <7* } с B* and  * n B * Ф 0 . Remark that Л * n B * is of type Gô in the subspace А* и B * . Since A*, B* are separated sets, it follows that according to Lemma 7 there exist decreasing sequences {A„},
{B„} o f open sets such that  * <= A„, B* <= B„ (for each n) and
(*) Ç )A „n Ç) Bn c zY \f ( L ' ) .
n n
Let fie a c p/ Then fie f ) A „ n f)B„. It is easy to see that {/!„}, {Bn}
0 ^ 0* n n
are infinite sequences. Moreover, we may assume that An Ф Am and B„ Ф Bm for n Ф m .
Write A^ = l ! n f ~ x(A*), B+ — t ! n f ~ i (B*) and D = FrtA^. Observe that D = FrL, В+ . Now, we shall prove that
(1) A# n D is a dense set in D.
Suppose, on the contrary, that there exist an element d e D and a neighbourhood Ud o f d such that f/d n D n Л * = 0 . Moreover, we may assume that q ^ U d (P\$Ud) if d Ф q (d Ф p x) . O f course, d e B*. Let e x , e 2 denote elements o f the arc L! such that d e b L { e x , e 2) <= Ud. We may addition
ally assume that e x Ф d Ф e 2 if q Ф d Ф p i . Observe that L t {d, e x) слА ^ф Ф or L v {d, e 2) r \ A ^ Ф 0 (if, for example, d = e x then we put L v {d, e x) = 0 ).
Suppose, for instance, that there exists an element d x e L L ( d, е х) п А ^ . O b viously d x ^ D , which means that d x e I n t L A # . O f course q Ф d x Ф p x .
Let h: I mü>L' be an arbitrary homeomorphism. Write h~1(A^.) = A h^ and h~1(dx) = dhx. Thus d^elntA^ and 0 Ф d x Ф\. Put
4 = inf \yh < d\: [ / ,
4 = sup {yh > dx : [ d { , / ] c ^ } .
Characterization o f Darboux functions 291
Let e1 = h(el), e2 = h(el). Then dx e L L {el , e2)\{ex, e2j <= A *. Now, we shall show that
(**) e2).
In fact, it is easy to see that d ^ L t (e1, e2)\ {e1, e2}. N ow we suppose that d = e1 (in the case o f d = e2 the proof is similar). Consider the arc L = L lJ(dl ,d). Thus £\{d} c and d e B*, which means that /(£ )==
{ / (d)} u Â*, where  * = f ( L ) n А* Ф 0 and /(d) eB*. Let c £\ {d } be an arbitrary net such that d elim z^.
^_____ lAef'
If { / (d)} n acp/ (z^) = 0 , then {w } quasi-cuts / (L) u acp/ (z^) between
0e¥' ^ 0e¥/
{/ (d )} and {/ (z *): ф e V} u acp / (гф) and {w } n f ( L ) = 0 , which is impos
te r
sible, because d is a Darboux point o f first kind o f /
N ow assume that {/ (d )} nacp/(z^,) Ф 0 . This means that f ( L ) ______ _ t e r
= { f ( d ) } u  * and {f(d)} n i * # 0 . According to Lemma 7 we infer that there exist decreasing sequences {A „ }, {B„} o f open sets such that { / ( d ) } c i „ and i * c B „ ( n = l , 2 , . . . ) and [ ] A n n f ] B n c Y \ f ( L ) . Ob-
П 9f
viously, / (d) e A„ n B„ (for each n), which contradicts the above inclusion.
This ends the proof o f (**).
According to (**) we deduce that either exe L v {d, dx) or e2eL,j(d, dx).
Suppose, for instance, that el e L t.(d, dx). Thus e1e D and e1e U d and conse
quently el e B*. Consider an arc L = L L {ex, dx). Note that ^ { e 1} c and in a similar way as above we can prove that el is not a Darboux point o f the first kind of /. The obtained contradiction ends the proof o f (1).
Similarly as (1), we can prove:
(2) В* n D is a dense set in D.
According to (1) and (2) we infer that D n (t'V fa ,
Now, we consider the set A x (sets A n and Bn are defined before the inclusion (*)). Observe that
Write ( D n f ~ l {Ax) n A A)\{q, px) Ф 0 .
<*i e ( Z ) n / '1(A1) n A !lc)\ {^ , p i}.
Then there exists an arc L x = L t (t1,tx), a1 eLx \{tx, and f ( L x) a A x.
Indeed, suppose, on the contrary, that such arc does not exist. Therefore there exists a net {у0}0еЛ c= L ' \ f ~ i ( A 1) such that ax elim y ^ we may
ôeà
assume that {у^Ьел c L L.(ax, y0J , for some S0 eA. Thus {w } quasi-cuts f (Lt,(oi1, yÔQ)) v acp f ( y 0) between { / ( » j ) } and { f { y ô): ô eA}u acp f ( y 0).
ôe A ôeA
On the other hand, {w } n / (L!) = 0 , which is impossible because a! is a Darboux point o f the first kind o f / The obtained contradiction ends the proof of the existence o f the arc L x for which <xl e L l \{tl , x l } and / ( L i ) a A x.
Now we shall consider the set Bx. According to (2) we deduce that there exists an element Pxe D n f ' 1 (Bx) n B^ tx}). In a similar way as above we may prove that there exists an arc L t = L , l ( f 1, £ 1) such that /?! eLx \ [t\, f j ) and f ( L 1) a B 1 (and consequently, f { L x) a A x n B x).
Suppose that for the sets A lf ..., A n- X; B lt ..., B„_x we have defined points al5 a „ _ i n such a way that
ai e(D r \ f ~ 1( A 1) п Л #)\ {q, px],
Ч Е Й n f ~ x{Ai) n A 1f n ( L i_ l \{ti- l , x i - l }) (/ = 2, n -1), PiED n f - ' t o ) n{Lt\ [ti, Т;|) (i = 1, л - 1)
and suitable arcs L x, Ц = L ii_ 1(ti, r j (i = 2, — 1) and Lï = L t .(fj-, £,) such that /(£*) c i j n (i = 1, n -1).
N ow we consider the set A„. According to (1), we can deduce that there exists an element ol„eD n f ~ i (Art) п Л * <^(Ln_ 1\{£„ _l5 £ „_ !}) and an arc
Ln = L L„_l (tn, О such that ctne L n\ [t„, xn] and f ( L J c A n.
Consider the set Bn. There exists an element fi„eD n n n(Lj,\ \t„, r„j) and f { L n)j=. B„ (and consequently f ( L„) eA „ nB„), where Ln Е^(£п, £„) and Pn e L n \1t n, £ „ , .
Continuing this procedure we obtain a decreasing sequence {Ln} of arcs
>4 00 00
such that Ln œ A „ n Bn for each n. O f course, П L„ Ф 0 . Let rjE П L„ c= LI.
n= 1 n=1
This means that f(r]) eA„ n B„ (n = 1 , 2 ,. . . ) and so f{rj) e 0 A„ n p B„ and
и л
consequently f ) A „ n f ) B nn f ( L ' ) = £ 0 , which contradicts (*). The obtained
П П
contradiction ends the proof of the theorem.
The next example shows that in the last theorem the necessary condition cannot be replaced by the condition “ every point x e l is a Darboux point of the second kind o f / ” .
Example 9. Let X = [0, 2] be a topological space with the natural topology. Let Y = [0, 1] и [2, 3] be a topological space with the topology generated by the neighbourhood system {B(y)}yeY, which is defined by the formula:
B(y) = \{y-\/n, y + 1 /л) n Y : n = ny, ny+ 1,
!(1 - 1/л, 1] u [2, 2+ 1/w): n = 1, 2, .
if y e Y\ \ l ) , if y = l,
where for every y e T \ J l ) , ny denotes a positive integer such that 1ф (y-\/ny, y+\/ny).
Let / : X -* ) be the transformation defined by the formula x if x e [0, 1),
x + 1 if xe[1, 2].
/ ( * ) =
Characterization o f Darboux functions 293 It is not hard to verify that every point from X is a Darboux point of the second kind of f but / is not a Darboux transformation.
In [20] T. Radakovic defined a new class of functions. The next definition presents a generalization of this notion to transformations whose domain and range are arbitrary topological spaces.
Definition 10. We say that f : X -* Y, where X, Y are arbitrary topo
logical spaces, is a Darboux transformation in the sense of Radakovic, if / ( / ) is a connected set for every arc / c A \
Theorem 11. Let f : X —> Y, where X and Y are arbitrary topological spaces. I f every element x e X is a Darboux point of the second kind of f then f is a Darboux transformation in the sense of Radakovic.
P r o o f. Suppose, on the contrary, that there exists an arc L = L ( u , v ) such that f ( l ) = A u B , where A, В are nonempty separated sets. Thus Л п б = 0 . Now, let us repeat word for word the part of the proof of Theorem 8 from “ ... W rite A — f ~ l (A )r \ L . . T to “ ... and let o* be an element of X such that xae L for each о ^ a* ...” .
Observe that (w| quasi-cuts f ( L ' ) u acp f ( x a) between \f(q)} and
fT > (T *
\f(xa): <7 ^ <7* Î u acp f { x j and n / (/ ') — 0 , which is impossible be-
<r> a*
cause q is a Darboux point of the second kind of /.
It is easy to see that there exists a Darboux function in the sense of Radakovic / : R —> R for which any point from R is not a Darboux point of the second kind o f /.
Example 9 shows that there exists the transformation / : X -* Y, where X is a T6-space such that Cf = X \ Jx0] and x 0 is a Darboux point of the second kind of f but / is not a Darboux transformation. Note that the space Y defined in this example is not even a Tx -space. The next theorem shows, among other things, that if the range o f a transformation possesing the above mentioned properties is a Tx-space, then / is a Darboux trans
formation.
Th e o r e m 12. Let f : X —> Y, where X is T2-space and Y is a Tx-space, be a function belonging to the class B 1 ( 1). Then f is Darboux transformation if and only if every point x e X is a Darboux point of the second kind of f.
P r o o f. Necessity follows from Theorem 8.
Sufficiency. First we shall show the following lemma:
P ) W e say that / e B 1 if C/jF Ф 0 for every closed set F с X .
(*) Let f : X Y, where X is an arbitrary topological space and Y is a Tx- space. I f x0 is a Darboux point of the second kind of f then f {x0) e / (/ \ UoD for every L = L ( x 0, a).
First, we assume that f ( L £ x 0, p)) = Y for each p e L \ { x 0}. Then there exists p0e L \ { x 0} such that f ( L t(x0, p0)) is a connected set and consequently f ( x 0) e f ( L L(x0, p0) ) \{/ (x 0)} c f ( L L{x0, p0)\ {x 0}).
Now, we consider the case where there exists an element p0e L\ |x0}
such that f ( L t(x0> p0)) Ф Y. Write L* = L t(x 0, p0). Let ae Y \ f ( L * ) and suppose, on the contrary, that f (x0) ф f (L\ {x0} ). It is easy to see that f (х0)ф / {L*\\x0} ) . Let { x , j î s ï c L * \ {x 0j be an arbitrary net, which con
verges to x 0. Then / ( x 0)<£acp/(xff). This means that (a) quasi-cuts / (L * )u a c p / (x ff) into !/ (x 0)] and f { L * \ {x 0}) u acp /(xff) and {a } n f ( L * )
ael ffGI
= 0 , which is impossible, because x0 is a Darboux point o f the second kind o f /. This ends the proof o f (*).
N ow we are going to the proof of our theorem.
Suppose, on the contrary, that / is not a Darboux transformation. Then there exists an arc L = L(u, v) с X such that f (L) = A kjB, where A, В are nonempty separated sets. Write A ' = f ~ 1( A ) n L, B' = f ~ 1( B ) n L and D
= Yrt A' = FrLB '. Then D is a nonempty closed set and so f\D has a point of continuity x 0. Assume, for instance, that x 0e A '. Then there exists an open (in У) set V such that f { x 0) e V and V n B = 0 . Let U be an arbitrary neighbourhood o f x 0 such that f ( U r \ D ) <= V. Note that
(1) U n D n B ' = 0 .
Let a and b denote the elements of L such that x0e L L{a, b) czU. W e may assume that а ф x 0 Ф Ь if и Ф x 0 Ф v. Let x t e L t (a, b ) n B ’. Write L
= Ll(x o, X i ) .
Let h: be a homeomorphism. Put A h = h~1(A' n t ) , Bh
= h~1( B ' n L ) . Suppose, for instance, that ■i = h~1(xi) (i = 0, 1). According to (1), x ^ I n t j B ' and so le ln t B * . Write <5 = in f{^ < 1: (/?, 1] c: Bh] and let a = h(S). It is easy to see that oteU n D . According to (*), we infer that афА' and consequently a e U r\D n B ', which contradicts (1). The obtained contra
diction ends the proof o f Theorem 12.
The next example shows that in the last theorem the necessary condition cannot be replaced by the condition “ every point x e X is a Darboux point of the third kind o f / ” .
Characterization o f Darboux functions 295
Example 13. Let X = / and let Y = [0, 1] u (i, 2] be a topological space with the topology generated by the neighbourhood system [B(y)}yeY, which is defined by the formula:
B(y) =
\U„(y) = ( y -l /и, y + l/ w )n [0 , i): и = 1 ,2 ,. . .} if y e [0 , l),
\ия(у )= ( \ - 1 / п , l ] u ( i i + l h ) : n = 2 , 3 , . . . } if y = J, '^ (у ) = [Ск-1/и, y+l/n) n (i, l ] ] u [ ( ( y + (n-l)/n,
у + (п+1)/и)\ [ y + l } ) n ( f , 2 ]]: n = 1, 2, ...}
if y e ( i , 1],
= (у-1 / и ,
y + l/n ) n ( f,
2]: и = 1 , 2 , . . . } ifye(f,
2].It is easy to see that Y is a Tx -space. Define a transformation / : X У by the formula / (x) = x. Since Cf = А Д Й }, we have / e # 1. It is not hard to verify that % is a Darboux point of the third kind of /. O f course, / is not a Darboux function.
The comments and Theorem 5.1 from [2 ] show that the notion o f a Darboux point is assumed in such a way that “ the local characterization” of a Darboux function takes place (see also [8], Property 1, p. 132). The next definition presents the notion of D-point, which is not equivalent to the notion o f a Darboux point in the sense o f A. Bruckner and J. Ceder (Example 15) but for which the local characterization o f a Darboux function takes place, too (Theorem 16).
Definition 14. Let /: X -> У, where X, Y are arbitrary topological spaces. We say that an element XqGA' is a D-point of f if x0 eCf or in the opposite case the following conditions are fulfilled:
1. If there exists a point p e X such that f ( L ( x 0, p)) = Y, then У is a connected space.
2. For an arbitrary arc L (x 0, a) there exists an element у e L (x 0,a)\\x0) such that for every z ЕЬЦхо>а)(х0, y), if К cuts Y between
!/ (x 0)! and ! / ( z ) \ then К n / ( / L(*0,a)(x 0, z)) Ф 0 .
Example 15. Let g0 : ( — oo, 0] u [1, + oo) —> R be a function defined by the formula
f ° if x e ( — oo, 0], g° {X) (sin2 if x e [ l , + o o ).
Let gn: [l/ (n + l), 1 /n)-+R (for n = 1, 2, ...) be a function defined in the following way:
7 — Prace M atem atyczne 27.2.
0„(x) = «
0 in the remaining cases.
Put / = V gn: R -» R . It is easy to see that 0 is a Darboux point o f / but it is not a D-point o f /.
It is not hard to verify that if / is an arbitrary transformation and x 0 is an arbitrary element from the domain of f then:
The other implications are not true.
Th e o r e m 16. Let f : X -* У, where X is an arbitrary topological space and Y is a Ts-space. Then f is a Darboux transformation if and only if every element x e X is a D-point of f
P r o o f. Necessity is obvious.
Sufficiency. Let us repeat word for word the first part o f the proof o f the sufficiency o f Theorem 8 (from “ ... Suppose on the contrary. . to .. Let
I * * } B ' n L L ( q, a) be a net such that q e W m x f ) .
Let U, V be open sets (in У) such that А c= U, В с V and U n V = 0 (see [5], Theorem 2.1.7, p. 97).
By у we denote an arbitrary element belonging to L t {q, a)\{q}. Now, we consider two cases:
(1*) Fr U Ф 0 or Fr УФ 0 . Suppose, for instance, that Fr U Ф 0 . O f course, Fr U cuts Y between A and B. Let o 0 denote an element of I such that xITQe L L(x0, y). Thus Fr U cuts У between {/ (x 0)} and {f ( xaQ)} and Fr U n / ( L ) = 0 , which is impossible because q is a D-point o f f
(2*) Fr U = 0 = Fr V. Then / (L) # У. Let c e Y \ f(L ). Put x0 is a Darboux point o f
the first kind o f /
x 0 is a Darboux point of the second kind o f /
x0 is a point o f continuity o f /
(I
x 0 is a D- point of /
x 0 is a Darboux point o f the third kind o f /
Characterization o f Darboux functions 297
(U \ { c } if c e U , ( Y\U if c e U ,
P = < y \ F if c e K ô = < F \ {c } if c e К It/ if с фи Kj V. ly \ (C / u {c }) if сф и Kj V.
Thus {c| cuts Y into P and Q, and consequently, it cuts Y between A and В and the proof is similar as the proof in case (1*).
W e shall close this paper with a theorem which presents some applica
tion o f the notion o f a Darboux point. Assume the following definitions: a transformation f : X -* Y we shall call closed (compact, connected) if / (К) is a closed (compact, connected) set, for every closed (compact, connected) set К in X.
V. L. Klee and W. R. Utz in [14] showed that if f : X -> Y (where X and Y are arbitrary matric spaces) is a connected and compact function and x0 is a local connectedness point o f X, then x0e C f . Generalizations of this result are contained in papers [9], [23], [24]. T. R. Hamlett in paper [10]
proved that if / is a closed, connected and monotone transformation defined on а Тъ-space and assuming their values in some compact T,-space, then / is a continuous transformation. H. Pawlak in [16] showed that if/ : Rn ->/?"' is a closed function, then / is continuous if and only if the image o f each segment is a connected set. Some results, which are connected with the characterization o f the continuity for closed function, are contained in J.
Jçdrzejewski’s habilitation thesis [13].
The next theorem is connected with the above mentioned problems.
Theorem 17. Let f : R n -> R m be a closed function. Then x0eCf if and only if x 0 is a Darboux point of the third kind of f
P r o o f. The necessity follows from Proposition 4.
Sufficiency. Suppose, on the contrary, that x0f C r . Then there exists
£ > 0 such that f ( K ( x0, 1 /n)) n ( R m\ K ( f (x0), e)) # 0 for each n = 1 ,2 ,...
Let [x„! be an arbitrary sequence such that x 0 = lim x„,
n -*00
f ( x „ ) $ K ( f ( x 0) , e ) ( n = \ ,2 ,...) and g (x 0, x„) > g (x 0, x j for n <m. By i'„ (n = 2 ,3 ,...) we denote a real number such that {>(x0, x„) < r„
< Q(x0, Хи-j). O f course, rn -> 0. Let L\ denote a segment with end-points at Xj and at some element o f F r( K ( x 0, r2)) and such that L\ n F r( K ( x 0, r2))
= { z } } . Let L\ denote a segment with end-points at x 2 and at some element of F r (K (x 0, r2)) and such that L\ c K ( x 0, r2) \ K ( x 0, r3). Then there exists an element z\ such that \z\\ = l \ n F r ( K ( x 0, r2)). If z\ = z j , then let /]
= lz}) and if z\ Ф z\, then let / ] denote an arc L{ z \, z\) c F r ( K ( x 0, r 2)). It is easy to see that L x = t j u £ j u L\ is some arc L ( x 1, x 2).
In a similar way we may define arcs: L 2 — L (x 2, x 3), L 3 = L (x 3, x4) etc.
00
Put L = {x 0} u U i^. Thus L is some arc.
i— 1
Let nl be a positive integer such that i £ + l/пг < e . Since x 0 is a Darboux point of the third kind o f /, we have
F r(k ( / (x0) , i e + l / n ) ) n / ( L ; (x0, x„)) ф 0 , for n ^ n t . Let
a „ e F r ( K ( / ( x 0), \e + \ / n ))n f (L l (x 0, x„)) (n ^ n j
and let y „ e f ~ l (ol„) n L t (x 0, x„) {n ^ л^. O f course, converges to x 0.
Let \yk \ be a subsequence o f \y„\?=ni such that ',akn! converges to a0 eFr(/C (/(x0), *-e)). Observe that ' ykn: n — 1, 2, ...] и !x0! is a closed set, b u t/C jb : n = 1,2, ... I ) is not a closed set. The obtained contra- diction ends the proof o f the theorem.
Note that in the last theorem the assumption “/ is a closed function”
can be replaced by the assumption “f is a closed function at x0” (see [19], Definition 3). O f course, the condition “ x 0 is a Darboux point of the third kind” can be replaced by the “ x 0 is a Darboux point o f the first kind” , “ x 0 is a Darboux point o f the second kind” or “ x 0 is the D-point of / ” .
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INSTYTUT M A TEM AT YK l UN IW ER SYTETU t O D Z K IE G O
INSTITt ТГ OF M \THF\1 \TICS. I O D Z I'NIYFRNITY. I Ô DZ. POI \ND
