1. Continuity of Darboux functions Nikita Shekutkovski, Beti Andonovic
UKIM, Skopje , Republic of Macedonia
Abstract
In 1965, Whyburn proved that if X is locally connected and first countable, Y is Hausdorff, and f: X→Y a function, then f is continuous iff f preserves compactness and connectedness. Definition: A function f: X → Y is preserving path-connectedness (a Darboux function) if the image of any path-connected subset of X is path connected. As an example, the derivative f ’ of a real differentiable function f defined on an interval is path-preserving although f ’ is not always continuous. In the paper we prove the following theorem:
Theorem: Suppose X is Hausdorff, locally path-connected and 1- countable, Y is Hausdorff, and f: X → Y a function. Then f is continuous iff f preserves compactness and f preserves path-connectedness.
Key words: function preserving path connectedness (Darboux function)

2. - Continuity of Darboux functions - Nikita Shekutkovski, Beti Andonovic
Definition. A function f : X → Y preserves path-connectedness (has a property of Darboux, or is a Darboux function), if the image of any path-connected subset of X is path-connected.
Remark. The derivative f’ of a real function f defined on an interval does not need to be continuous. However, the derivative f’ has the property of Darboux, i.e. if d is a real number such that f(a) < d < f(b), then there exists a real number c between a and b such that f(c) = d.

3. - Continuity of Darboux functions - Nikita Shekutkovski, Beti Andonovic
Example. A function having a Darboux property and is not continuous, is the function of Cesaro
ω : [0, 1] → [0, 1] defined by
for 0 ≤ x ≤ 1, and x = 0, a1... an is a dyadic expression of x (if x has two expressions we take the finite expression, i.e. with infinite number of zeros).

4. First we will show the following:
- Continuity of Darboux functions - Nikita Shekutkovski, Beti Andonovic
Next we want to show that the following conditions ) and 2’) are also equivalent:
1) f is continuous.
2’) Image of a compact set is compact and f preserves path-connectedness.
It is clear that from 2) it follows 2’). We will need to show that from 2’) it follows 2).
First we will show the following:
Theorem 1. Let f : X → Y be a mapping and X be locally path-connected metric space. If f maps compact to compact and preserves path connectedness, then the image of any subset of X that is both compact and connected, is connected (and also compact).

5. Abstract
In 1965, Whyburn proved that if X is locally connected and first countable, Y is Hausdorff, and f: X→Y a function, then f is continuous iff f preserves compactness and connectedness. Definition: A function f: X → Y is preserving path-connectedness (a Darboux function) if the image of any path-connected subset of X is path connected. As an example, the derivative f ’ of a real differentiable function f defined on an interval is path-preserving although f ’ is not always continuous. In the paper we prove the following theorem:
Theorem: Suppose X is Hausdorff, locally path-connected and 1- countable, Y is Hausdorff, and f: X → Y a function. Then f is continuous iff f preserves compactness and f preserves path-connectedness. 1 2

6. First we will show the following:
- Continuity of Darboux functions - Nikita Shekutkovski, Beti Andonovic
Next we want to show that the following conditions ) and 2’) are also equivalent:
1) f is continuous.
2’) Image of a compact set is compact and f preserves path-connectedness.
It is clear that from 2) it follows 2’). We will need to show that from 2’) it follows 2).
First we will show the following:
Theorem 1. Let f : X → Y be a mapping and X be locally path-connected metric space. If f maps compact to compact and preserves path connectedness, then the image of any subset of X that is both compact and connected, is connected (and also compact).

7. To prove Theorem 1, we will need the following
- Continuity of Darboux functions - Nikita Shekutkovski, Beti Andonovic
To prove Theorem 1, we will need the following
Proposition 1. Let (Cn) be a sequence of compact, path-connected sets, such that each is a subset of the previous one. Then
is also compact.
(Note. Proposition 1 also holds for connected spaces instead of path-connected spaces.)

8. - Continuity of Darboux functions - Nikita Shekutkovski, Beti Andonovic
Example. As an example we consider the space consisting of two parallel lines in the plane, and of a spiral a(t) starting from a point a(0) between the two lines and a(t) approaching to both lines as t → ∞ . Then is a decreasing sequence of connected sets while their intersection is a union of two parallel lines, i.e., their intersection isn’t connected (see picture on the right).

9. Proof of Proposition 1. C is compact as an intersection of compacta.
- Continuity of Darboux functions - Nikita Shekutkovski, Beti Andonovic
Proof of Proposition 1. C is compact as an intersection of compacta.
Let us assume that C = A U B, where A and B are separated. For two points a  A and b  B, there exists a path kn : I → Cn , such that kn(0) =a, kn(1) = b .
If we put , then
We define a mapping for each n, hn : kn(I) → R in the following way:

10. - Continuity of Darboux functions - Nikita Shekutkovski, Beti Andonovic
Since for each n, hn(a) < 0, hn(b) > 0, there exists zn  kn (I), so that
hn (zn) = 0, i.e., d(zn, A) = d(zn, B), where zn = kn (tn), 0 < tn < 1.
There exists a convergent subsequence (znk), such that znk → z  Z’ A U B.
Because of d(z, A) = d(z, B), it follows z A and z B, which is a contradiction.

11. - Continuity of Darboux functions - Nikita Shekutkovski, Beti Andonovic
Proof of Theorem 1. Let f : X → Y be a mapping and let C be both connected and compact in X. For each x  C, there exists a sequence of path-connected open neighbourhood , each a subset of the previous one, so that:
Since
is compact, it follows that there exists a finite union that covers C.
Let

12. - Continuity of Darboux functions - Nikita Shekutkovski, Beti Andonovic
There exists a subsequence (Un) of (Wn), so that
It follows that ,
and
Then, f(C) is compact and also connected as an intersection of compact path-connected sets, each a subset of the previous one.

13. - Continuity of Darboux functions - Nikita Shekutkovski, Beti Andonovic
Proposition 2. Let X be a Hausdorff space and f : [0, 1]→X preserves path connectedness and compactness. Then f is continuous. Proof. Let C be connected in [0, 1]. It follows that C is an interval, which means C is path-connected. f preserves path-connectedness, so it follows that f(C) is path-connected, so f(C) is connected. It follows that f is continuous. Proposition 3. Let h : [0, 1] → X be an embedding (i.e. h([0, 1]) is an arc) and let f : X → Y be a function that preserves path-connectedness and compactness. Then, the restriction f|h(I) is continuous. Proof. Let C h(I) be connected. Then C is a homeomorphic image of an interval of I, and because of Proposition 2, it follows the conclusion in Proposition 3.

14. - Continuity of Darboux functions - Nikita Shekutkovski, Beti Andonovic
Theorem 2. Let X be Hausdorff, locally path-connected and 1-countable, Y be Hausdorff, and let f : X → Y be a function preserving path-connectedness and compactness. Then f is continuous. (Note: Y does not have to be locally connected.) Proof. First, since X is locally path-connected, it is arcwise connected. ([1]) Let z  X. We will show that f is continuous by showing that the sequence of images of the members of an arbitrary sequence that converges to z, would converge to f(z)  Y .

15. - Continuity of Darboux functions - Nikita Shekutkovski, Beti Andonovic
Let (an) be a sequence, such that an → z and let {Vk : k  N} be a family of open arcwise connected neighbourhoods of z, each a subset of the previous one. We may choose an increasing sequence (nk) of natural numbers (except in the trivial case), and a subsequence (ank) of (an), so that ank  Vnk , while ank−1 Vnk . The set {ank : k  N} U {z} is compact, and since f : X → Y maps compact to compact, it follows that is compact.

16. - Continuity of Darboux functions - Nikita Shekutkovski, Beti Andonovic
By taking subsequences we may assume that f(ank) converges. If f(ank) → f(z) then the proof is completed. Let us assume that f(ank) → ω, where ω ŧ f(z). Then, because of the compactness of the set {f(ank) : k  N} U {f(z)}, it follows that ω  {f(ank) : k  N}, i.e. ω = f(aj). Now, because of the properties of the neighbourhoods Vnk, there exists an arc Ak from ank to z .

17. - Continuity of Darboux functions - Nikita Shekutkovski, Beti Andonovic
We can make the construction of the arcs Ak by induction, in the following way: For A1, A2, ..., Ak−1 , we put A = A1 U A2 U ... U Ak−1. If A*k is an arbitrary arc in Vnk from ank to z, we define t* = inf{t|hk(t)  A}, where h*k : I → A*k is an isomorphism and we put bnk = h*k (t*). Then we define the arc Ak from ank to bnk to be as same as the arc A*k , and from bnk to z to be identical with some of the previously constructed arcs. We define a set , which is compact because of the compactness of the arcs and the way how neighbourhoods Vk are chosen.

18. - Continuity of Darboux functions - Nikita Shekutkovski, Beti Andonovic
Now, let C be an arbitrary connected set in Q. 1) If C intersects more than one arc, then in the intersection of any two such arcs and C there is contained some bnk. 2) If we consider the case when C intersects only one arc, then it follows that Cn = C ∩ hn(I) is homeomorphic to an interval. Because of 1) and 2) it follows that f(C) is connected. It follows the restriction f|Q: Q → Y is continuous. So, if W is a neighbourhood of f(z), such that f(aj) W, it follows that there exists a neighbourhood U of z, such that f|Q (U ∩ Q) W. There exists k’, such that for each k > k’, ank  U ∩ Q, and so f(ank)  W, which is a contradiction to f(ank) → f(aj).

19. - Continuity of Darboux functions - Nikita Shekutkovski,Beti Andonovic
Definition. The map f : X → Y has a removable discontinuity at x0 if there exists a y0  Y, such that for any neighbourhood V of y0 there exists a neighbourhood U of x0 such that f(U\{x0}) V. Theorem 3. Let X be locally path-connected and 1-countable and let f : X → Y be a function preserving path-connectedness and compactness. Then f is continuous at x0 if and only if it has at worst a removable discontinuity at x0 .

20. - Continuity of Darboux functions - Nikita Shekutkovski,Beti Andonovic
REFERENCES.
[1] R. Engelking, General Topology, Warszawa, 1977
[2] J.Gerlits, I. Juhasz, L. Soukup, Z. Szemiklossy, Characterizing continuity by preserving compactness and connectedness, Top. Appl. 138 (2004), pp
[3] McMillan, E., R., (1970) On continuity conditions for functions, Pacific Journal of Mathematics, 32 (2),
[4] N.Shekutkovski, G.Markoski, B.Andonovik’, Functions preserving connectedness or path-connectedness, Mat. Bilten 31(LXVII), (2007), 5-12
[5] G.T Whyburn, Continuity of Multifunctions, Proceedings N.A.S Vol 154, (1965), pp – 1501