Discontinuity points of a function with a closed and connected graph
aa r X i v : . [ m a t h . GN ] J un DISCONTINUITY POINTS OF A FUNCTION WITH A CLOSEDAND CONNECTED GRAPH
MICHAŁ STANISŁAW WÓJCIK
Abstract.
The main result of this paper states that for a function f : R → Y with a closed, connected and locally connected graph, where Y is a locally com-pact, second-countable metrisable space, the graph over discontinuity pointsremains locally connected. Motivation.
It is a classic result that for a function f : X → Y with a closedgraph and Y Hausdorff, a sufficient and necessary condition for being continuousis sub-continuity [Fuller 68, 3.4]. However, it is an interesting question how, forvarious spaces X and Y , additional topological properties of a closed graph arerelated to the continuity of a function. It is known that, for a function f : R → R with a closed graph, a sufficient and necessary condition for being continuous is theconnectedness of the graph [Burgess 90]. In 2001, Michał R. Wójcik and I statedthe question whether this result can be extended to f : R → R [Wójcik 2004, 9]– this problem was then propagated by Cz. Ryll-Nardzewski. The answer to thisquestion is negative. The first known discontinuous function f : R → R with aconnected and closed graph was shown by J. Jel´ınek in [Jel´ınek 2003]. It can beshown that the graph of Jel´ınek’s function in not locally-connected [Mrwphd 2008,A4]. Therefore a new question was stated, whether connectedness together withthe local connectedness of the graph is a sufficient and necessary condition of beingcontinuous for a function f : R → R with a closed graph. This question, as far as Iknow, remains open. In this paper, I show some properties of the set of discontinuityof a function f : R → R with a closed, connected and locally connected graph,hoping that they might be useful in the main research. Result.
The main result of this paper states that for a function f : R → Y witha closed, connected and locally connected graph, where Y is a locally compact,second-countable metrisable space, the graph over discontinuity points remains lo-cally connected. This result is given as Corollary 16 as a consequence of some deeptopological properties of the real plane and the more generic Theorem 13.1. Notation and terminology
Definition 1.
Let
X, Y be topological spaces and f : X → Y be an arbitraryfunction.(1) We will denote by C ( f ) or C f the set of all points of continuity,(2) by D ( f ) or D f – set of all points of discontinuity.(3) We will denote by π a projection operator π : X × Y → X and π ( x, y ) = x .(4) For A ⊂ X , by f | A we will denote a restriction of f to the subdomain A . In the context of function f : X → Y , we will not use a separate symbol todenote the graph of f , for f itself, in terms of Set Theory, is a graph. So whenwe use Set Theory operations and relations with respect to f , they should beunderstood as operations and relations with respect to the graph. Whenever thisnaming convention might be confusing, we will add the word “graph”, e.g. “ f hasa closed graph”. Definition 2.
Let Y be a topological space and y n ∈ Y be an arbitrary net. Wewill write y n → ∅ or lim y n = ∅ iff y n has no convergent subnet. Functions with a closed graph
It will be helpful to cite two well-known theorems concerning functions with aclosed graph:
Theorem 3. If X is a topological space, Y is a compact space, f : X → Y and thegraph of f is closed, then f is continuous. (for proof: e.g. [Wójcik 2004, T2]) Theorem 4. If X is a Bair and Hausdorff space, Y is a σ -locally compact space, f : X → Y and the graph of f is closed, then C(f ) is an open and dense subset of X . (for proof: e.g. [Doboˇs 85, T2])3. Graph over discontinuity points
I will begin with several well-known facts:
Fact 5.
Every non-empty metrisable compact space is a continuous image of theCantor set. (for proof e.g. [Engelking 89, 4.5.9])
Fact 6. If X is a connected and locally arcwise-connected metrisable space, F is aclosed subset of X , C ⊂ [0 , is the Cantor set and f : C → F is continuous and f ( C ) = X , then f has a continuous extension f ∗ : [0 , → X . (for proof: e.g. [Kuratowski II 66, 50.I.5]) Fact 7. If X is a topological space and A, B ⊂ X are both closed and locallyconnected, then A ∪ B is locally connected. (for proof: e.g. [Kuratowski II 66, 49.I.3]) Fact 8. If X is a compact and locally connected space, Y is a Hausdorff space, f : X → Y is continuous and f ( X ) = Y , then Y is compact and locally connected. (for proof: notice that since X is compact and Y is Hausdorff, f is a quotientmapping and local connectedness is invariant under quotient mappings [Whyburn 52,T2]) Lemma 9. If X is a connected metrisable space, and F is a connected and closedsubset, X \ F = A ∪ B , where Clo ( A ) ∩ B = Clo ( B ) ∩ A = ∅ , then F ∪ A is connectedand closed.Proof. [Kuratowski II 66, 46.II.4] (cid:3) ISCONTINUITY POINTS OF A FUNCTION WITH A CLOSED AND CONNECTED GRAPH3
Lemma 10. If X is a locally connected metrisable space, F is a locally connectedand closed subset and S is a sum of some connected components of X \ F , then S ∪ F is locally connected.Proof. [Kuratowski II 66, 49.II.11] (cid:3) Theorem 11. If X is a connected and locally connected, locally compact, second-countable metrisable space, E is a continuum in X and U is an arbitrary openneighbourhood of E , then there exists a locally connected continuum F such that E ⊂ F ⊂ U and X \ F has finitely many connected components.Proof. Since X is locally compact, Hausdorff and locally connected, there existsan open neighbourhood V x of the point x such that Clo ( V x ) is a continuum and Clo ( V x ) ⊂ U for each x ∈ E . Since E is compact, there exist x , x , . . . , x n ∈ E such that E ⊂ n S i =1 V x i . Let V = n S i =1 V x i . Notice that Clo ( V ) is compact and E ⊂ V ⊂ Clo ( V ) ⊂ U . By Fact 5, there is a continuous function f : C → E , where C is the Cantor set and f ( C ) = E . Since X is metrisable and locally compact, itis completely metrisable and therefore, by the Mazurkiewicz-Moore theorem, X islocally arcwise-connected, and thus V is locally arcwise-connected. V is connectedby construction. Therefore by Fact 6 there is a function f ∗ that is a continuousextension of f such that f ∗ : [0 , → V . Let F = f ∗ ([0 , . By Fact 8, F isa locally connected continuum and E ⊂ F ⊂ V . Let S V be a family of all theconnected components of X \ F that are subsets of V . Let S ∞ be a family of allthe other connected components of X \ F . Notice that, due to the connectedness of S , (1) S ∩ ∂V = ∅ for any S ∈ S ∞ . Since X is locally connected, all the connectedcomponents of X \ F are open in X . Since ∂V ⊂ S S ∞ , by virtue of (1) and thecompactness of ∂V , the family S ∞ is finite. Let F = F ∪ S S V . By Lemma 9, F is connected and closed. Since F ⊂ Clo ( V ) , F is a continuum. By Lemma 10, F is locally connected. Obviously, E ⊂ F ⊂ V ⊂ U . Since X \ F = S S ∞ and S ∞ isfinite, the proof is complete. (cid:3) Lemma 12. If X is a Hausdorff space, Y is a topological space, f : X → Y is afunction with a closed graph, D = D ( f ) , E is a compact subset of the graph and U is a relatively open subset of the graph such that U ⊂ E ⊂ f , then U ∩ f | D = U ∩ f | ∂π ( E ) . Moreover, if ( x, f ( x )) ∈ U ∩ f | D and X \ π ( E ) ∋ x n → x , then f ( x n ) → ∅ .Proof. Since E is compact, by Theorem 3, f | π ( E ) is continuous. If x ∈ Int ( E ) , then f is continuous in x , so x D . Therefore U ∩ f | D ⊂ U ∩ f | ∂π ( E ) is obvious. Wewill show inverse inclusion by contradiction. Assume that ( x, f ( x )) ∈ U ∩ f | ∂π ( E ) and x is a continuity point of f . Since π ( E ) is compact and X is Hausdorff, π ( E ) is closed, so x ∈ π ( E ) . Notice that ( x, f ( x )) ∈ U , so by the continuity of f atpoint x , there is an open neighbourhood V of x , such that f | V ⊂ U . But U ⊂ E ,so x ∈ V ⊂ π ( E ) . This contradicts how x was chosen. Now we will show the“moreover part”. Take any ( x, f ( x )) ∈ U ∩ f | D and X \ π ( E ) ∋ x n → x . Since U ⊂ E , ( x n , f ( x n )) U . So no subnet of f ( x n ) is convergent to f ( x ) . But thegraph of f is closed, so f ( x n ) → ∅ . (cid:3) Theorem 13. If X is a connected and locally connected, locally compact, second-countable metrisable space, Y is a locally compact, second-countable metrisable MICHAŁ STANISŁAW WÓJCIK space, f : X → Y , D = D ( f ) and the graph of f is closed, connected and lo-cally connected, then for each x ∈ D there is an open in the graph topology U anda locally connected continuum E such that(1) ( x, f ( x )) ∈ U ⊂ E ⊂ f ,(2) U ∩ f | D = U ∩ f | ∂π ( E ) (so x ∈ ∂π ( E ) ),(3) if X \ π ( E ) ∋ x n → x , then f ( x n ) → ∅ ,(4) X \ π ( E ) has finitely many connected components.Proof. Notice that f is a connected and locally connected, locally compact, second-countable metrisable subspace of X × Y . Take an arbitrary x ∈ D . Choose openin the graph topology set U , such that ( x, f ( x )) ∈ U and Clo f ( U ) is a continuum.By Theorem 11, there exists a locally connected continuum E ⊂ f such that f \ E has finitely many connected components and U ⊂ E . By Lemma 12, f | D ∩ U = f | ∂p ( E ) ∩ U and for any subsequence X \ π ( E ) ∋ x n → x we have f ( x n ) → ∅ . ByFact 8, p ( E ) is a locally connected continuum and since f \ E has finitely manyconnected components and π is continuous, the set π ( f \ E ) = π ( f ) \ π ( E ) = X \ π ( E ) also has finitely many connected components. (cid:3) Theorem 13 has an interesting consequence for X = R , namely ∂π ( E ) from theabove theorem is locally connected, which implies (by Theorem 8) that f | D has alocally connected graph. To prove this, let me refer to the following theorem: Theorem 14. If A is a locally connected continuum in R , and S is a connectedcomponent of R \ A , then ∂S is a locally connected continuum.Proof. Since R is homeomorphic with a unit sphere without one point, it’s enoughto apply [Kuratowski II 66, 61.II.4]. (cid:3) Let’s formulate a simple consequence of the above.
Theorem 15. If A is a locally connected continuum in R and R \ A has finitelymany connected components, then ∂A is locally connected.Proof. Let S , S , . . . S n be connected components of R \ A . S , S , . . . S n areopen, since R \ E is an open subset of a locally connected space and thus locallyconnected. Since we’re dealing only with a finite number of open sets, the belowequation holds. ∂A = ∂ ( R \ A ) = ∂ ( n [ i =1 S i ) = n [ i =1 ∂S i . By Theorem 14, ∂S i is locally connected for i = 1 , , . . . , n . Therefore, by Fact 7, ∂A is locally connected. (cid:3) By applying Theorem 13, Theorem 3 and Theorem 8, we immediately get thefollowing corollary.
Corollary 16. If Y is a locally compact, second-countable metrisable space, f : R → Y has a closed, connected and locally connected graph, then f | D f has alocally connected graph. One might propose that as the local connectedness of f | D f is a local property, itmight be enough to assume only local connectedness of f . Unfortunately, there isa simple example that shows that the connectedness of f is necessary in Corollary16 and Theorem 13. ISCONTINUITY POINTS OF A FUNCTION WITH A CLOSED AND CONNECTED GRAPH5
Example 17.
Let r n = n ( n +1) , B n = { ( x, y ) ∈ R : q x + ( y − n ) < r n } . f ( x, y ) = for y ≥ and ( x, y ) S ∞ n =1 B n , y for y < ,n + tan( π r n q x + ( y − n ) ) for ( x, y ) ∈ B n for n = 1 , . . . Note that in the above example B n is a sequence of pairwise disjoint open discsconvergent to the point (0 , . f = 0 on the whole half plane R × [0 , ∞ ) exceptdiscs B n . f ≥ n on B n and converges to infinity on ∂B n . Therefore, it is easy tonotice that the graph of f is closed and not connected. The local connectedness ofthe graph is obvious everywhere except the point (0 , , . But as f ≥ n on B n and f = 0 on R × [0 , ∞ ) \ S ∞ n =1 B n , it’s enough to see that R × [0 , ∞ ) \ S ∞ n =1 B n is locallyconnected at the point (0 , . Thus the graph of f is locally connected. However, f | D f = ( R ×{ }∪ S ∞ n =1 ∂B n ) ×{ } and is not locally connected in (0 , , . It’s alsoeasy to notice that for any open in the graph topology set U such that (0 , , ∈ U and for any locally connected continuum E such that U ⊂ E ⊂ f , R \ π ( E ) hasinfinitely many connected components, since B n ∩ π ( E ) = ∅ and ∂B n ⊂ E foralmost all n . References [Burgess 90] C. E. Burgess,
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