Real Analysis Exchange Vol. 25(2), 1999/2000, pp. 579–598

Krzysztof Ciesielski^{∗}, Department of Mathematics, West Virginia University, Morgantown, WV 26506-6310. e-mail: K Cies@math.wvu.edu

web page: http://www.math.wvu.edu/homepages/kcies

Harvey Rosen, Department of Mathematics, University of Alabama, Tuscaloosa, AL 35487. e-mail: hrosen@gp.as.ua.edu

TWO EXAMPLES CONCERNING EXTENDABLE AND ALMOST CONTINUOUS FUNCTIONS

Abstract

The main purpose of this paper is to describe two examples. The first is that of an almost continuous, Baire class two, non-extendable function f: [0, 1] → [0, 1] with a G_{δ} graph. This answers a question of Gibson [15]. The second example is that of a connectivity function F : ℝ^{2} → ℝ with dense graph such that F^{−1}(0) is contained in a countable union of straight lines. This easily implies the existence of an extendable function f: ℝ → ℝ with dense graph such that f^{−1}(0) is countable. We also give a sufficient condition for a Darboux function f: [0, 1] → [0, 1] with a G_{δ} graph whose closure is bilaterally dense in itself to be quasi-continuous and extendable.

1 Definitions and Notation

Our terminology is standard and follows [6]. We consider only real-valued functions of one or more real variables. No distinction is made between a function and its graph. A restriction of a function f: X → Y to a set A ⊂ X is denoted by f ↾ A. By ℝ and ℚ we denote the set of all real and rational numbers, respectively, while I will stand for the interval [0, 1]. The closure of a set A ⊂ ℝ^{n} is denoted by cl(A), its boundary by bd(A) and its diameter by diam(A). The first coordinate projection of a set A ⊂ ℝ^{2} will be denoted by pr(A).

Key Words: extendable functions, peripherally continuous functions.

Mathematical Reviews subject classification: Primary: 26A15; Secondary: 54A25.

Received by the editors March 19, 1999

∗The work of the first author was partially supported by NSF Cooperative Research Grant INT-9600548 with its Polish part financed by Polish Academy of Science PAN.

Papers authored or co-authored by a Contributing Editor are managed by a Managing Editor or one of the other Contributing Editors.

The ordinal numbers will be identified with the sets of all their predecessors and cardinals with the initial ordinals. In particular 2 = {0, 1} and the first infinite ordinal ω number is equal to the set of all natural numbers {0, 1, 2, … }.

We will also use the following terminology [16]. For X ⊆ ℝ^{n} a function f: X → ℝ is:

• Darboux if f[K] is a connected subset of ℝ (i.e., an interval) for every connected subset K of X;

• almost continuous (in the sense of Stallings) if each open subset of X ×ℝ containing f also contains a continuous function from X to ℝ [27];

• connectivity function if the graph of f ↾ Z is connected in Z × ℝ for any connected subset Z of X;

• extendable function if there is a connectivity function F: X × [0, 1] → ℝ such that f(x) = F (x, 0) for every x ∈ X;

• peripherally continuous if for every x ∈ X and for all pairs of open sets U and V containing x and f(x), respectively, there exists an open subset W of U such that x ∈ W and f[bd(W)] ⊂ V.

The classes of these functions are denoted by D, AC, Conn, Ext and PC, respectively, where the space X will be always clear from the context. Recall also (see e.g. [16] or [8]) that for the functions from ℝ to ℝ and from I into I we have the following strict inclusions

Ext ⊊ AC ⊊ Conn ⊊ D ⊊ PC  (1)

while in the Baire class one all these classes coincide. (See Brown, Humke and Laczkovich [5].) On the other hand, for the classes of functions from ℝ^{n}, with n > 1, into ℝ we have

[figure: Ext = Conn = PC; AC ∩ D; AC; D] where arrows denote strict inclusions. (Equation Ext = Conn on ℝ^{n} is proved in [10].)

2 Almost Continuous Non-Extendable Function with G_{δ} Graph

As noted above the classes listed in (1) cannot be distinguished within the family B_{1} of Baire one functions. On the other hand, the classes of functions on ℝ or I listed in (1) can be distinguished within the family B_{2} of Baire class two functions. (See e.g. [8, thm 1.2].) The next natural question is: for which classes G containing B_{1} but not B_{2} will the inclusions remain strict? For several classes G strictly between B_{1} and B_{2} this has been investigated by Brown [3] and Brown, Humke and Laczkovich [5].

In this section we will investigate the inclusions in (1) within the class G = G_{δ} ∩ B_{2}, where G_{δ} is the class of functions with G_{δ} graphs. Notice that B_{1} ⊂ G_{δ} since for every pointwise limit f of continuous functions f_{n}: I → ℝ the graph of f is equal to ⋂_{n=1}^{∞} ⋃_{i=n}^{∞}{⟨x, y⟩: |f_{i}(x) − y| < 2^{−n}}. Clearly also every function f from G_{δ} is Borel, since for every open U ⊂ ℝ the set f^{−1}(U) = pr(f ∩ [I × U]) = I \ pr(f ∩ [I × (ℝ \ U)]) is simultaneously analytic and coanalytic, and hence Borel. (See e.g. [23, p. 489] or [19, p. 89].) On the other hand there are functions in G_{δ} of arbitrary high Borel class. This follows from the fact that every Borel set is a one-to-one continuous image of a closed subset of ℝ \ ℚ (see e.g. [19, 15.3, p. 89]) which is clearly G_{δ} in ℝ. Indeed, if A ⊂ I is of the class at least α for some α < ω_{1}, take one-to-one continuous functions f_{0} from a closed subset of (0, 1) \ ℚ onto A and f_{1} from a closed subset of (2, 3) \ ℚ onto I \ A. Then f = f_{0}^{−1} ∪ f_{1}^{−1} : I → ℝ has a G_{δ} graph and is of Borel class at least α, since f^{−1}[(0, 1)] = A. Thus, if B_{α} stands for the Baire class α functions, then we have the following relations for every 2 ≤ α < ω_{1}, where arrows denote strict inclusions.

[figure: B_{1}; G_{δ} ∩ B_{2}; B_{α}; G_{δ}; Borel]

The properness of inclusions in (1) within the class G_{δ} has been summarized by Brown in [4]. (See also [16, sec. 3].) In particular, he noticed that the inclusions AC ⊂ Conn ⊂ D ⊂ PC remain proper within this class G_{δ} leaving open the problem of properness of inclusion Ext ⊂ AC in the G_{δ} class. (This problem is stated explicitly by Gibson in [15, Question 4].) In fact the examples exhibiting properness of the inclusions AC ⊂ Conn ⊂ D ⊂ PC within the class G_{δ} ∩B_{2} can be found in the literature. For AC ⊂ Conn Jones and Thomas [18] (also see Thomas [28]) constructed a non almost continuous function f: I → I with a connected G_{δ} graph. This f is also B_{2}, but it is not pointed out in the paper. Also Brown [3, remark 3] gives a bit stronger example and states explicitly that it is in G_{δ} ∩ B_{2} ∩ Conn but not in AC. The properness of the inclusion Conn ⊂ D within G_{δ}∩B_{2} is stated explicitly by Brown in [3, example 1 and remark 3], using the example constructed by him in [1, example 2]. (Also, in [24] Miller gives an example of a Darboux function f: I → I having a G_{δ} graph but no fixed point and so the graph of f is not connected. This function is also B_{2}, but it is not pointed out in the paper.) Concerning the properness of the inclusion D ⊂ PC in G_{δ} ∩ B_{2} Brown [4] states that there is one (not mentioning B_{2}) and Miller [24] describes a peripherally continuous non Darboux function with G_{δ} graph, which turns out to be also in B_{2}.

The main goal of this section is to describe an example justifying properness of Ext ⊂ AC within G_{δ} ∩ B_{2}, thus answering a question of Gibson in [15, Question 4]. However, we will notice also that the small modifications of this example justify also properness of the remaining inclusions of (1) within G_{δ} ∩ B_{2}.

[figure: 1; 1; c_{1}; d_{1}; \frac{1}{2}; a_{1}; b_{1}; a_{2}; b_{2}; \frac{3}{4}; c_{2}; d_{2}; c_{3}; d_{3}; \frac{7}{8}; c_{4}; d_{4}; \frac{15}{16}; c_{5}; d_{5}]

Figure 1: function f ∈ G_{δ} ∩ B_{2} ∩ AC \ Ext from Example 2.1 Example 2.1. There exists an almost continuous, non-extendable, Baire class two function f: I → I with G_{δ} graph.

Proof. The example is a slight modification of a function constructed by Ciesielski and Jastrzȩbski in [8, thm 3.1]. It also slightly simplifies the argument given there.

Let C be the Cantor ternary set in I and let J be the family of all component intervals of I \ C. For i ∈ {0, 1} let J_{i} be the family of all intervals J ∈ J of length 3^{−2n+i} for some integer n. Thus, {J_{0}, J_{1}} is a partition of J. Let {(a_{n}, b_{n}): 0 < n < ω} and {(c_{n}, d_{n}): 0 < n < ω} be the enumerations of J_{0} and J_{1}, respectively. Define function f: [0, 1] → [0, 1] in the following way. (See Figure 1.)

• For every 0 < n < ω we put f(a_{n}) = f(d_{n}) = 0, f(b_{n}) = f(c_{n}) = 1−2^{−n} and extend it linearly on each interval [a_{n}, b_{n}] and [c_{n}, d_{n}]. We refer to b_{n} and c_{n} as upper endpoints for f of [a_{n}, b_{n}] and [c_{n}, d_{n}], respectively.

• For all other x’s we put f(x) = 0.

(Ciesielski and Jastrzȩbski define their function by assigning to f(b_{n}) and f(c_{n}) value 1 instead of 1 − 2^{−n}. It does not have G_{δ} graph since f ∩ (I × {1}) = ⋃_{n=1}^{∞}{b_{n}, c_{n}} × {1} is not G_{δ} being countable dense in C × {1}.) First note that the graph of f is a G_{δ} set as a union of three G_{δ} sets:

• f ↾ (C \ ⋃_{n=1}^{∞}{b_{n}, c_{n}}) = (C \ ⋃_{n=1}^{∞}{b_{n}, c_{n}}) × {0},

• f ↾ (I \ C) = ⋂_{n=1}^{∞}{⟨x, y⟩: x ∈ I \ C & |f(x) − y| < 2^{−n}} and

• f ↾ ⋃_{n=1}^{∞}{b_{n}, c_{n}} = ⋃_{n=1}^{∞}{⟨b_{n}, 1 − 2^{−n}⟩, ⟨c_{n}, 1 − 2^{−n}⟩} which is discrete.

Also, since the first two restrictions are continuous, it is easy to see that the preimage f^{−1}(U) of any open set U ⊂ I is a union of a G_{δ}-set and an F_{σ}-set; so f is of Baire class two.

Next we will show that f is not extendable. By way of contradiction, assume that f is extendable; that is, that there is a connectivity function F: I^{2} → I with F (x, 0) = f(x) for all x ∈ I. Thus F is peripherally continuous. We will deduce from this that there exists an x ∈ I such that F (x, 0) = 1, which implies f(x) = 1, contradicting our definition of f. For this we define inductively the sequences ⟨p_{1}, p_{2}, p_{3}, …⟩ of upper endpoints for f and ⟨B_{0}, B_{1}, B_{2}, …⟩ of closed connected subsets of I^{2} such that for each n < ω we have

F [B_{n}] ⊂ (1 − 2^{−n+1}, 1], B_{n} ∩ B_{n+1} ≠ ∅, diam(B_{n}) ≤ 2^{−n},

and

⟨p_{n+1}, 0⟩ ∈ B_{n}, F (p_{n+1}, 0) = f(p_{n+1}) ≥ 1 − 2^{−(n+1)}.

We will start with B_{0} = [c_{2}, d_{2}] × {0} and p_{1} = c_{2}. Clearly, the above conditions are satisfied. So assume that p_{n} and B_{n−1} satisfying the above are already constructed for some n > 0. Then p_{n} is the upper endpoint for f of some interval J_{0} ∈ J. We have to find B_{n} and p_{n+1}. So, choose an ε > 0 less than the length of J_{0} such that ε < min{2^{−(n+1)}, p_{n}, 1 − p_{n}, diam(B_{n−1})} and

(∗) if J ∈ J \ {J_{0}} is closer to p_{n} than ε, then it has the length at most 2^{−(n+1)} and f(p) ≥ 1 − 2^{−(n+1)} for the upper endpoint p for f of J.

Since F (p_{n}, 0) ≥ 1−2^{−n} and F is peripherally continuous we can find an open neighborhood W_{n} ⊂ [0, 1]^{2} of ⟨p_{n}, 0⟩ with diameter less than ε and such that F [bd(W_{n})] ⊂ (1−2^{−n+1}, 1]. Without loss of generality we can also assume that bd(W_{n}) is connected. (This is a standard argument. See e.g. [27].) Note that the choice of ε guarantees that bd(W_{n})∩B_{n−1} ≠ ∅ ≠ bd(W_{n})∩(I_{0}×{0}), where I_{0} is a component of I \J_{0} containing p_{n}. This is so, since bd(W_{n}) disconnects [0, 1]^{2}, while B_{n−1} and I_{0} × {0} are connected, containing ⟨p_{n}, 0⟩ ∈ W_{n} and of diameter greater than ε ≥ diam(bd(W_{n})). Let z ∈ I_{0} be such that ⟨z, 0⟩ ∈ bd(W_{n}) ∩ (I_{0} × {0}). Since F (z, 0) ∈ F [bd(W_{n})] ⊂ (1 − 2^{−n+1}, 1] ⊂ (0, 1], there exists a J ∈ J such that z ∈ cl(J). Let p_{n+1} be the upper endpoint for f of J. Then, by (∗), f(p_{n+1}) ≥ 1 − 2^{−(n+1)} and the distance of ⟨p_{n+1}, 0⟩ from ⟨z, 0⟩ ∈ bd(W_{n}) is at most 2^{−(n+1)}. The set B_{n} is defined as a union of bd(W_{n}) and a closed segment joining ⟨z, 0⟩ and ⟨p_{n+1}, 0⟩. It is easy to see that the inductive conditions are satisfied. This finishes the construction.

Now, to finish the argument notice that for every n < ω the set B^{n} = ⋃_{k=n}^{∞} B_{k} is connected, has diameter at most ∑_{k=n}^{∞} 2^{−k} = 2^{−n+1} and contains ⟨p_{n+1}, 0⟩. In particular, there exists p ∈ I with p = lim_{n} p_{n}. We claim that F (p, 0) = 1. Indeed, if this is not the case, then there exists an n < ω such that |F (p, 0) − 1| > 2^{−n+2}. Take ε > 0 less than the diameter of B^{n} and let U be an open neighborhood of ⟨p, 0⟩ of diameter less than ε and such that F [bd(U)] ⊂ (F (p, 0) − 2^{−n+1}, F (p, 0) + 2^{−n+1}). Then there exists a point w ∈ B^{n}∩bd(U), since U ∩B^{n} ≠ ∅, U has diameter less than diam(B^{n}) and B^{n} is connected. But then F (w) belongs to both (F (p, 0)−2^{−n+1}, F (p, 0)+2^{−n+1}) and F [B^{n}] ⊂ (1 − 2^{−n+1}, 1], which is impossible, since these two sets are disjoint. So, F (p, 0) = 1. But this is impossible as well since then we would have f(p) = F (p, 0) = 1 and f does not attain 1. This contradiction finishes the proof that f is not extendable.

To show that f is almost continuous let G be an open subset of I^{2} containing the graph of f. Notice that for every x ∈ [0, 1] there exists an interval (r_{x}, s_{x}) (for x = 0 and x = 1 we consider intervals [r_{x}, s_{x}) and (r_{x}, s_{x}], respectively) such that • x ∈ (r_{x}, s_{x}), f(r_{x}) = f(s_{x}) = 0 and

• there is a continuous function g_{x}: [0, 1] → ℝ with g_{x} ↾ [r_{x}, s_{x}] ⊂ G and such that g_{x}(t) = 0 for t ∉ (r_{x}, s_{x}).

Indeed, if f(x) = 0, then it is easy to find r_{x} < s_{x} for which g_{x} ≡ 0 works. If f(x) ≠ 0, then x ∈ cl(J) for some J ∈ J. For the sake of simplicity assume that f is increasing on J; that is, that J ∈ J_{0}, the other case being similar. Let J = (a, b). Then f(b) > 0. Let U ⊂ G be an open circular neighborhood of ⟨b, f(b)⟩. Then there exists an interval J′ = (c, d) ∈ J_{1} with b < c such that f ↾ (c, d) intersects U. Let u ∈ (c, d) be such that ⟨u, f(u)⟩ ∈ U. Put r_{x} = a, s_{x} = d, define g_{x}(b) = f(b), g_{x}(u) = f(u), g_{x}(0) = g_{x}(r_{x}) = g_{x}(s_{x}) = g_{x}(1) = 0 and extend g_{x} linearly on each interval with these endpoints. Thus g_{x} has a “hat” shape. We have g_{x} ↾ [r_{x}, s_{x}] ⊂ G since g_{x} ↾ [b, u] ⊂ U ⊂ G and g_{x} ↾ ([a, b] ∪ [u, d]) = f ↾ ([a, b] ∪ [u, d]) ⊂ G. Now choose a finite subcover {[r_{0}, s_{0}), (r_{1}, s_{1}]} ∪ {(r_{x_{i}}, s_{x_{i}}): i ≤ n}, of the cover {[r_{0}, s_{0}), (r_{1}, s_{1}]} ∪ {(r_{x}, s_{x}): x ∈ (0, 1)} of the interval [0, 1]. Then

g(x) = max{g_{0}(x), g_{1}(x), g_{x_{0}}(x), g_{x_{1}}(x), …, g_{x_{n}}(x)}

is continuous and g ⊂ G. This ends the proof that f is almost continuous.

Note that as in [8, Proposition 3.4] one can show the following.

Remark 2.1. If f is from Example 2.1, then there exists a characteristic function χ_{B} of a meager F_{σ} subset B of C such that f_{0} = f +χ_{B} is extendable.

The next example is a slight modification of the function from Example 2.1 and is a variant of examples of Jastrzȩbski [17] and Kellum [21]. In what follows we will use the notation from Example 2.1.

Example 2.2. There exists a connectivity, non almost continuous, Baire class two function f: I → I with G_{δ} graph.

Proof. Define f as follows. (See Figure 2.)

• For every 0 < n < ω we put f(a_{n}) = f(c_{n}) = 0, f(b_{n}) = f(d_{n}) = 1−2^{−n} and extend it linearly on each interval [a_{n}, b_{n}] and [c_{n}, d_{n}]. (Of course we could now reduce our enumeration of J to just one sequence, instead of two.)

• For all other x’s we put f(x) = 0.

The proof that such f is in G_{δ} ∩B_{2} is identical to that for f from Example 2.1. The proof that f is connectivity is easy and is essentially the same as for the examples in [17, 21]. The fact that f ∉ AC follows from [21, Lemma 1].

[figure: 1; 1; c_{1}; d_{1}; \frac{1}{2}; a_{1}; b_{1}; a_{2}; b_{2}; \frac{3}{4}; c_{2}; d_{2}; c_{3}; d_{3}; \frac{7}{8}; c_{4}; d_{4}; \frac{15}{16}; c_{5}; d_{5}]

Figure 2: function f ∈ G_{δ} ∩ B_{2} ∩ Conn \ AC from Example 2.2

Example 2.3. There exists a Darboux, non connectivity, Baire class two function f: I → I with G_{δ} graph.

Proof. This is a slight modification of function k: I → I constructed by Ciesielski and Kellum in [9]. Let C denote the Cantor middle two-fifths set:

C = \{∑_{n=1}^{∞} \frac{i_{n}}{5^{n}} : i_{n} ∈ \{0, 2, 4\} for every n\} .

Geometrically, C is obtained from I by first removing the pair of intervals (1/5,2/5) and (3/5,4/5) from I; then by removing similar pairs of intervals (the middle two fifths) from each of [0,1/5], [2/5,3/5] and [4/5,1], etc. Let {⟨(a_{n}, b_{n}), (c_{n}, d_{n})⟩: n < ω} be an enumeration of all such pairs (with b_{n} < c_{n}). Also, put C^{∘} = I \ ⋃_{n<ω}[a_{n}, b_{n}] ∪ [c_{n}, d_{n}] and let Δ denote the diagonal {⟨x, x⟩: x ∈ I} in I^{2}. Define f as follows.

• For n < ω put f(c_{n}) = 0, choose f(b_{n}) > d_{n} > b_{n} from (1 − 2^{−n}, 1), pick a_{n} < f(a_{n}) < f(d_{n}) < d_{n} and extend f linearly on the intervals [a_{n}, b_{n}] and [c_{n}, d_{n}]. Note that f[[a_{n}, b_{n}] ∪ [c_{n}, d_{n}]] = [0, f(b_{n})] and that f ↾ ⋃_{n<ω}[a_{n}, b_{n}] ∪ [c_{n}, d_{n}] is disjoint from Δ.

• Put f(0) = 1/2 and f(x) = 0 for x ∈ C^{∘} \ {0}.

The proof that such an f is in G_{δ} ∩B_{2} is identical to that used in Example 2.1. The function f is not connectivity, since Δ separates its graph. It is Darboux, since f[J] = [0, 1) for every interval J intersecting C^{∘}.

Example 2.4. There exists a peripherally continuous, non Darboux, Baire class two function f: I → I with G_{δ} graph.

Proof. We use here the notation from Example 2.3. Define f as follows.

• For n < ω put f(a_{n}) = 0, f(b_{n}) = \frac{1}{2} − 2^{−n−3}, f(c_{n}) = \frac{1}{2} + 2^{−n−3}, f(d_{n}) = 1−2^{−n−3} and extend f linearly on intervals [a_{n}, b_{n}] and [c_{n}, d_{n}].

• Put f(x) = 0 for all x ∈ C^{∘}.

The proof that such an f is in G_{δ} ∩ B_{2} is essentially identical to that used in Example 2.1. The function f is in PC \ D since f[J] = [0, 1) \ \{\frac{1}{2}\} for every interval J intersecting C^{∘}.

Corollary 2.1. The inclusions

Ext ⊂ AC ⊂ Conn ⊂ D ⊂ PC

remain strict for the Baire class two functions f: I → I with G_{δ} graphs.

3 An Extendable Function with Dense Graph and Countable Zero Level

In this section we concentrate on functions from ℝ to ℝ, though essentially all presented results remain the same for functions from I to I. Our main result here is the following.

Theorem 3.1. There exists an extendable function f: ℝ → ℝ with dense graph for which f^{−1}(0) is countable.

The motivation for searching for such an example comes from several directions. First of all in 1970 Brown [2] proved that if F is the class of connectivity functions from ℝ to ℝ, then (A) if f ∈ F has a dense graph, then every nowhere dense set N is f-negligible

for F; that is, any function g: ℝ → ℝ equal to f outside of N remains in F;

(B) part (A) is false for countable sets N; that is, there exists an f ∈ F with a dense graph and a countable dense set D ⊂ ℝ which is not f-negligible for F.

The similar result for the class F of almost continuous functions has been proved in 1982 by Kellum [20]. Also for the class F of extendable functions in 1994 Rosen [25] proved part (A). However, so far there has been no example justifying (B) for the class Ext. The function f from Theorem 3.1 clearly justifies (B) for F = Ext, since the countable zero level D = f^{−1}(0) cannot be f-negligible for Ext. (Modifying f on D we can get a non Darboux function.)

The other motivation comes for the following theorem of Ciesielski and Rosłanowski [12, thm. 3.1], (which has been used in constructing an additive, almost continuous, non extendable function with the SCIVP property).

Proposition 3.2. If f: ℝ → ℝ is extendable and has a dense graph, then f has the following (super SCIVP) property

For every x, y ∈ ℝ and for each perfect K between f(x) and f(y) there is a perfect C between x and y such that f[C] ⊂ K and f ↾ C is continuous

     strictly monotone.

It seems natural to ask whether in the above the phrase “strictly monotone” can be replaced with “constant” or, more generally, the perfect set K be replaced by a singleton. We do not know the answer to the first of these questions. However the example from Theorem 3.1 is a counterexample for the second conjecture.

Theorem 3.1 is also related to the following open problem [16, Question 9.31]. Is it true that g: ℝ → ℝ is a product of two extendable functions if and only if g has a zero in each subinterval in which it changes sign? (This characterization is true if the class of extendable functions is replaced by the class of Darboux, connectivity, or almost continuous functions.) It gives the positive answer in the following particular case.

Corollary 3.3. If g: ℝ → ℝ is such that g^{−1}(0) is dense in ℝ, then g is a product of two extendable functions.

Proof. The proof is a slight modification of the fact that every function g: ℝ → ℝ is a sum of two extendable functions. (See [25] or [11].)

Let f be from Theorem 3.1. Then there exists a dense G_{δ} subset G of ℝ which is f-negligible for Ext. (See [25, Theorem 1].) Also there exists a homeomorphism h_{0} of ℝ such that G ∪ h_{0}[G] = ℝ. Moreover, f_{0} = f ∘ h_{0}^{−1} is extendable and h_{0}[G] is f_{0}-negligible for Ext.

Let h be a homeomorphism of ℝ such that h[f^{−1}(0) ∪ f_{0}^{−1}(0)] ⊂ g^{−1}(0) and put f_{1} = f ∘ h^{−1} and f_{2} = f_{0} ∘ h^{−1} = f ∘ h_{0}^{−1} ∘ h^{−1}. Then, f_{1} and f_{2} are extendable and the sets G_{1} = h[G] and G_{2} = h[h_{0}[G]] are f_{1}- and f_{2}-negligible, respectively. Also, Z = f_{1}^{−1}(0) ∪ f_{2}^{−1}(0) ⊂ g^{−1}(0). Define

f̂_{1}(x) = \{ \begin{matrix}\frac{g(x)}{f_{2}(x)} & for x ∈ G_{1} \ Z \\ f_{1}(x) & otherwise \end{matrix} and f̂_{2}(x) = \{ \begin{matrix}f_{2}(x) & for x ∈ G_{1} ∪ Z \\ \frac{g(x)}{f_{1}(x)} & otherwise. \end{matrix}

Then f̂_{1} and f̂_{2} are well defined since f_{1} and f_{2} are not 0 outside Z. They are extendable, since they are respective modifications of f_{1} and f_{2} on negligible sets G_{1} \ Z ⊂ G_{1} and ℝ \ (G_{1} ∪ Z) ⊂ G_{2}. Also,

• for x ∈ Z we have f̂_{1}(x)f̂_{2}(x) = 0 = g(x),

• for x ∈ G_{1} \ Z we have f̂_{1}(x)f̂_{2}(x) = \frac{g(x)}{f_{2}(x)} f_{2}(x) = g(x) and

• for x ∈ ℝ \ (G_{1} ∪ Z) we have f̂_{1}(x)f̂_{2}(x) = f_{1}(x) \frac{g(x)}{f_{1}(x)} = g(x).

Thus, g = f̂_{1}f̂_{2}.

Theorem 3.1 will be concluded from the following theorem. The construction is a modification of Ciesielski-Recław’s construction from [11, Theorem 3.3]. However, the triangles in the triangulations are not equilateral, as in the generalization of [11, Theorem 3.3] presented in [13, prop. 2.3].

Theorem 3.4. There exists a connectivity function F: ℝ^{2} → ℝ with graph dense in ℝ^{3} such that F^{−1}(0) is contained in a countable union of straight lines in ℝ^{2}.

Before we prove Theorem 3.4 let us notice how it implies Theorem 3.1.

Proof of Theorem 3.1. Let F be as in Theorem 3.4. Note that by rotating the domain ℝ^{2} of F we obtain the same kind of example. Thus, without loss of generality we can assume the countable family L of lines with F^{−1}(0) ⊂ ⋃ L does not contain a line parallel to the x-axis. Then every line L_{y} = ℝ × {y} intersects F^{−1}(0) in a countable set. Moreover, by examining the function F constructed below (or by an argument similar to that of Rosen from [25, Theorem 1]) there exists a dense G_{δ} subset G of ℝ^{2} such that G is F -negligible for the class Conn. Thus, by the Kuratowski-Ulam theorem (a category analog of the Fubini theorem), there is y ∈ ℝ such that G ∩ L_{y} is a dense G_{δ} subset of L_{y}. This implies that the graph of F ↾ L_{y} is dense in L_{y} × ℝ. It is easy to see that f defined by f(x) = F (x, y) has all the desired properties.

Proof of Theorem 3.4. The constructed function F will be peripherally continuous, so connectivity. (See e.g. [16, Theorem 8.1] or [7, p. 171].)

Basic Idea: By induction on n < ω we will construct a sequence ⟨S_{n}: n < ω⟩ of triangular “grids,” that is, triangulations of the plane. We will begin with S_{0} formed with equilateral triangles of side length 1 as in Figure 3. Then, at the stage n + 1, we will subdivide each triangle from S_{n} into finitely many pieces making sure that their sizes tend to zero. The grid S_{n} will be identified with the points on the edges of triangles forming it and we will assume that S_{n} ⊆ S_{n+1} for all n < ω. With each grid S_{n} we will associate a continuous function f_{n}: S_{n} → ℝ which is linear on each side of a triangle from S_{n}. Moreover, each f_{n+1} will be an extension of f_{n}. The function F will be defined as an extension of ⋃_{n<ω} f_{n} such that F^{−1}(0) ⊂ ⋃_{n<ω} S_{n}. Thus, F^{−1}(0) will be contained in a countable union of straight lines; those that contain sides of triangles from S_{n}’s.

[figure]

Figure 3: grid S_{0}

Terminology: In what follows a triangle will be identified with the set of points of its interior or its boundary. For a grid S we say that a triangle T is from S if the interior of T is equal to a component of ℝ^{2} \ S.

For a triangle T its basic partition will be its split into ten triangles as in Figure 4. The central triangle T̂ of Figure 4 will be referred to as the middle quarter of T. It is similar to T and its sides are parallel to the sides of T and of 1/4 of their respective lengths. Also, T and T̂ have the same center and the vertices on the sides of T are at their centers. It is easy to see that the diameter of each of triangle from the basic partition of T is at most half of the diameter of T. Also T̂ ∩ bd(T) = ∅.

[figure]

Figure 4: the middle quarter T̂ of a triangle T and the basic partition of T

If a function f is defined on the three vertices of a triangle T, its basic extension is defined as the unique function f̂: T → ℝ extending f whose graph is a subset of a plane in ℝ^{3}. Notice, that f̂ is linear on each side of the triangle T and that f̂ extends f even if the function f has already been defined on some side of T as long as f is linear on this side. In particular, if f is defined on a grid S and is linear on each side of every triangle from S, then f̂: ℝ^{2} → ℝ is defined in a natural way. Inductive construction: We will inductively define two increasing sequences: ⟨S_{n}: n < ω⟩ of triangular grids and ⟨f_{n} ∈ ℝ^{S_{n}}: n < ω⟩ of continuous functions such that the following inductive conditions are satisfied for every n < ω.

(i) For each triangle T from S_{n} function f_{n} on T is linear not identically 0 and f_{n}[T] is a subset of one of the intervals: [−2^{n}, −1], [−1, 0], [0, 1], or [1, 2^{n}].

(ii) The side length of each triangle from S_{n} is at most 1/2^{n}.

(iii) The variation of f_{n} on each triangle from S_{n} is at most 1/2^{n}.

(iv) If n > 0, k ∈ {−1, 1}, then for every triangle T from S_{n−1} for which (k · f_{n})[T] ⊂ [0, ∞) the following holds.

(a) With every non-zero dyadic number i/2^{n} ∈ [−2^{n}, 2^{n}], where i belongs to D_{n} = {i ∈ ℤ: − 4^{n} ≤ i ≤ 4^{n} & i ≠ 0}, we associate a triangle T_{i} ⊆ T̂ from S_{n} such that all triangles T_{i} are disjoint,

f̂_{n} [T̂ \ ⋃_{i∈D_{n}} T_{i}] = \{k\} and f̂_{n} \left[{ \atop T̂_{i}}\right] = \{i/2^{n}\} for every i ∈ D_{n}.

(b) For every x ∈ T \ T̂ either f̂_{n−1}(x) ≤ f̂_{n}(x) ≤ k or f̂_{n−1}(x) ≥ f̂_{n}(x) ≥ k.

To start the induction define grid S_{0} as in Figure 3 with all sides of length 1 and let f_{0} : S_{0} → ℝ be identically 1. It is easy to see that conditions (i)–(iv) are satisfied for such a choice.

Next, assume that for some n > 0 we already have S_{n−1} and f_{n−1} satisfying (i)–(iv). We will define S_{n} and extend f_{n−1} to f_{n}: S_{n} → ℝ such that (i)–(iv) will still hold. Let T be a triangle from S_{n−1} and put k = 1 if f̂_{n−1}[T] ⊂ [0, ∞) and k = −1 for f̂_{n−1}[T] ⊂ (−∞, 0]. The S_{n}-triangulation of T will be a refinement of the basic triangulation of T. This will guarantee (ii).

Clearly f_{n} is already defined on bd(T). We define f_{n} on each vertex of the middle quarter T̂ of T by assigning it the value k. Next, on each triangle T′ from the basic partition of T except for T̂ (see Figure 4) we extend f_{n} to f̂_{n} linearly. Such an extension is unique since f_{n} is already defined on each vertex of T′. Note that this and the choice of k guarantee (iv)(b). This also gives us (i) for any triangle from a subtriangulation of T′.

[figure]

Figure 5: some triangles T_{i} of the grid of T̂

Thus, f̂_{n} is defined so far on T \ T̂. To extend f̂_{n} to T̂ we proceed as follows. Partition T̂ into a grid S such that S contains 2 · 4^{n} disjoint triangles {T_{i}: i ∈ D_{n}}. (See Figure 5.) We extend f̂_{n} to T̂ \ ⋃_{i∈D_{n}} T_{i} by assigning it the constant value k, while on each T̂_{i} with i ∈ D_{n}, f̂_{n} is identically i/2^{n}. This guarantees the reminder of condition (iv) as well as (i) for the triangles considered so far. Finally, f̂_{n} is extended linearly on each triangle from the basic partition of T_{i} using the fact that it is already defined on each of its vertices. This finishes the construction of f̂_{n}.

Now, to make sure that (iii) and the remaining part of (i) are satisfied we will refine the triangulations defined so far to the grid S_{n}. This will lead to the definition of f_{n} as f̂_{n} ↾ S_{n}.

For (i) we proceed as follows. If k and i ∈ D_{n} have the same sign, then f̂^{n} ↾ T_{i} does not attain value 0 and (i) is already satisfied by the triangles from the basic partition of T_{i}. In this case S_{n} on T_{i} will be formed from the triangles considered so far. On the other hand, if k and i ∈ D_{n} have different signs, then (f̂^{n} ↾ T_{i})^{−1}(0) is a boundary of a triangle T′ similar to T_{i}. Then we subtriangulate the basic partition of T_{i} in such a way that every edge of T′ is covered by the edges of this subtriangulation and that each triangle from the subtriangulation intersects at most one edge (excluding the end points) of T′. If, in addition, (f̂^{n} ↾ T_{i})^{−1}(−k) is nonempty, then it is a boundary of a triangle (or the entire triangle) and we will make sure that this boundary is a subset of our subtriangulation.

Condition (iii) is easily guaranteed by refining the triangulation described so far. This finishes the inductive construction.

Now, F is defined on S = ⋃_{n<ω} S_{n} as f = ⋃_{n<ω} f_{n}. We will extend it to ℝ^{2} making sure that F (x) ≠ 0 for every x ∈ ℝ^{2} \ S. This will guarantee that F^{−1}(0) ⊂ S. Note also that, by (iv)(a), the graph of f is already dense in ℝ^{3}. So any extension F of f will have this property as well.

To extend F to ℝ^{2} pick an x ∈ ℝ^{2} \ S. Note that, by (i), f̂_{n}(x) ≠ 0 for every n < ω. For every n < ω let T_{n}^{x} be the triangle from S_{n} containing x and let N = {n < ω: x ∈ T̂_{n}^{x}}. We consider two cases.

Case 1. The set N is infinite. If N^{k} = {n ∈ N: f̂_{n}[T̂_{n}^{x}] = {k}} for k ∈ {−1, 1}, then N = N^{−1} ∪ N^{1} and so at least one of the sets N^{−1} and N^{1} is infinite. Let k = 1 if N^{1} is infinite and put k = −1 otherwise. In this case we define F (x) = k. Notice that this guarantees that F is peripherally continuous at x since for any n ∈ N^{k}

F [bd(T̂_{n}^{x})] = f_{n}[bd(T̂_{n}^{x})] = {k} = {F (x)},

x belongs to the interior of T̂_{n}^{x} and the diameter of T̂_{n}^{x} is at most 1/2^{n}. Case 2. The set N is finite. Fix an m < ω such that N ⊂ {0, …, m−2}. Then x ∈ T_{n}^{x} \ T̂_{n}^{x} for every n ≥ m. So, if k ∈ {−1, 1} is such that (k · f_{m})[T_{m}^{x}] ⊂ [0, ∞), then, by (iv)(b), either f̂_{m}(x) ≤ f̂_{m+1}(x) ≤ f̂_{m+2}(x) ≤ ⋯ ≤ k or f̂_{m}(x) ≥ f̂_{m+1}(x) ≥ f̂_{m+2}(x) ≥ ⋯ ≥ k. So the limit L = lim_{m→∞} f̂_{m}(x) is well defined and not equal to 0. We put F (x) = L. This, together with (ii) and (iii), guarantees that F is peripherally continuous at x.

This finishes the construction of function F and the argument that F is peripherally continuous on ℝ^{2} \ S. To see that F is peripherally continuous on S take an x ∈ S. Then, there exists a k < ω such that x ∈ S_{n} for every n ≥ k. For any such n let T_{n} be the set of all triangles from S_{n} to which x belongs. Then x belongs to the interior of the polygon P_{n} = ⋃ T_{n}. Moreover, by (ii) and (iii), the variation on the boundary of P_{n} and the diameter of P_{n} are at most 2/2^{n}. So, the sequence ⟨P_{n}⟩ guarantees that F is peripherally continuous at x. This finishes the proof of Theorem 3.4.

4 Quasi-Continuous Extendable Functions

To state the results of this section we need the following additional terminology and facts.

For x ∈ I let l_{x} = {x} × I and for function f: I → I let C(f) stand for the set of points of continuity of f. Recall that for a Darboux function f: I → I the set cl(f) ∩ l_{x} is connected for every x ∈ I and that C(f) is a dense G_{δ} provided f has a G_{δ} graph. (See e.g. [18].) The function f: I → I is quasi-continuous if f ↾ C(f) is dense in the graph of f. A function f: I → I is said to have a closure that is bilaterally dense in itself if cl(f ↾ (0, x)) ∩ l_{x} = cl(f ↾ (x, 1)) ∩ l_{x} for each x ∈ (0, 1).

Croft’s function f: I → I from [14] is Darboux, upper semicontinuous (hence of Baire class one) and 0 almost everywhere, but is not identically 0. It follows that the closure of f is bilaterally dense in itself. However, since it is not quasi-continuous, Croft’s function does not satisfy the second category condition in the following result. But a function like Kellum and Garrett’s [22, Example 1] does satisfy it and the rest of the hypothesis.

Theorem 4.1. Suppose f: I → I is a Darboux function with a G_{δ} graph whose closure is bilaterally dense in itself. Also suppose for every point ⟨x, f(x)⟩ of f there exists an open neighborhood W ⊂ I^{2} of ⟨x, f(x)⟩ such that if B is closed and nowhere dense in pr(f ∩ W), then pr^{−1}(B) ∩ (f ∩ W) is nowhere dense in f ∩ W. Then f is quasi-continuous and extendable.

Proof. Let A = {x ∈ I: cl(f) ∩ l_{x} is nondegenerate}. Assume f is not quasi-continuous at some point x_{0} ∈ I. Then there exists a rectangular open neighborhood W = I_{1} × I_{2} ⊂ I^{2} of ⟨x_{0}, f(x_{0})⟩ that obeys the second category condition of the hypothesis and that contains no point of f ↾ C(f). Note that pr(f ∩ W) = {x ∈ A ∩ I_{1} : ⟨x, f(x)⟩ ∈ W }. According to the Alexandroff theorem, the G_{δ} subset f ∩ W of I^{2} is homeomorphic to a complete metric space. Therefore, by hypothesis, pr(f ∩ W) is of second category in itself.

Observe that if x ∈ pr(f ∩ W), then cl(f) ∩ l_{x} is a nondegenerate interval meeting I^{2}\W and so cl(f ∩W)∩l_{x} contains a nondegenerate interval containing ⟨x, f(x)⟩. To show f ∩W is nowhere dense in pr(f ∩W)×I let [a, b]×[c, d] be a rectangle in I^{2} that meets pr(f ∩ W) × I and let G_{1} ⊃ G_{2} ⊃ ⋯ be open subsets of I^{2} such that f ∩ W = (⋂_{n=1}^{∞} G_{n}) ∩ (pr(f ∩ W) × I). For all positive integers n and rational numbers r and s with c ≤ r < s ≤ d let H(n, r, s) be the set of all x ∈ pr(f ∩ W) ∩ [a, b] for which some component of l_{x} \ G_{n} meets both I ×{r} and I ×{s}. Each H(n, r, s) is closed in pr(f ∩W)∩[a, b] and each point of pr(f ∩ W) ∩ [a, b] belongs to some H(n, r, s). Therefore there exist n, r, s and an interval (u, v) such that H(n, r, s) ⊃ pr(f ∩ W) ∩ (u, v) ≠ ∅. Since the sets f ∩ W and [pr(f ∩ W) ∩ (u, v)] × (r, s) are disjoint, f ∩ W is nowhere dense in pr(f ∩ W) × I.

For each positive integer n let Q_{n} be the set of all x ∈ pr(f ∩ W) for which cl(f ∩ W) ∩ l_{x} contains an interval containing ⟨x, f(x)⟩ of length at least \frac{1}{n}. Each Q_{n} is closed in pr(f ∩ W). Since f ∩ W is nowhere dense in pr(f ∩ W) × I, each Q_{n} is nowhere dense in pr(f ∩ W). Because pr(f ∩ W) is of second category and pr(f ∩ W) = ⋃_{n=1}^{∞} Q_{n}, this a contradiction. By [26, Theorem 1] a Darboux quasi-continuous function f whose closure is bilaterally dense in itself is extendable.

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