146 (1994)

**Connectivity of diagonal products of Baire one functions**

### by

**A. M a l i s z e w s k i (Bydgoszcz)**

**Abstract. We characterize those Baire one functions f for which the diagonal product** *x 7→ (f (x), g(x)) has a connected graph whenever g is approximately continuous or is a* derivative.

**Abstract. We characterize those Baire one functions f for which the diagonal product**

**I. Introduction. It is well known for a long time that the product of two** derivatives need not be a derivative. Moreover, the characteristic function of every closed set can be written as the product of two bounded derivatives [2].

### So the graph of the product of two derivatives need not be connected, whence the graph of the diagonal product of two derivatives need not be connected.

### On the other hand, it is well known (and easy to prove) that the product of a bounded approximately continuous function with a bounded deriva- tive is a derivative again, so its graph is connected. However, the graph of the diagonal product of a bounded approximately continuous function with a bounded derivative is not necessarily connected. In 1963 Neugebauer [7]

*constructed a bounded approximately continuous function f and a bounded* *derivative g such that f (0) = g(0) = 0 and [f (x)]* ^{2} *+ [g(x)]* ^{2} *≥ 1/4 for x 6= 0,* *so the graph of f 4g is not connected. In this paper our aim is to character-* *ize those Baire one functions f for which the diagonal product f 4g has a* *connected graph whenever g is approximately continuous or is a derivative.*

**II. Preliminaries. The real line (−∞, ∞) is denoted by R and the set** of positive integers by N. The only measure used is Lebesgue measure in R *and all integrals are Lebesgue integrals. For each set A ⊂ R, int A denotes* *its (Euclidean) interior, cl A its closure, fr A its boundary and |A| its outer* measure.

**II. Preliminaries. The real line (−∞, ∞) is denoted by R and the set**

*1991 Mathematics Subject Classification: Primary 26A15; Secondary 54C08.*

*Key words and phrases: Darboux function, peripheral continuity, approximate conti-* nuity, derivative, Baire one function, diagonal product.

### Supported by a KBN Research Grant 2 1144 91 01, 1992–94.

[21]

*The word function means a mapping from R into R unless otherwise* *explicitly stated. The graph of a function f : X → Y will be denoted* *by Γ (f ). We denote by D the family of all derivatives. If X and Y are* *metric spaces, then the family of all Baire one functions from X into Y* (i.e., pointwise limits of sequences of continuous functions) will be denoted *by B* _{1} *(X, Y ). We write B* _{1} *for B* _{1} *(R, R).*

*The terms d-closed, d-interior (d-int) etc. will refer to the Denjoy topol-* *ogy (density topology) on R. (See, e.g., [3], [9], [5].) We say that a function f* *is approximately continuous if it is continuous relative to the Denjoy topol-* ogy. The family of all approximately continuous functions will be denoted *by C* _{ap} *. Recall that each element of C* _{ap} is a Baire one function and each *bounded element of C* ap is a derivative (see, e.g., [1]).

*We denote by b the family of all bounded functions. We drop the “∩”*

*sign between classes of functions, e.g., bC* ap denotes the family of all bounded approximately continuous functions.

*We denote by C the family of all continuous functions, by C* *a.e.* the family *of all functions which are continuous almost everywhere and by C* _{0} the family of all functions which are continuous except possibly at one point. It is *well known that elements of C* ap *C* *a.e.* are exactly those functions which are continuous with respect to the so-called a.e.-topology [8]. Recall that a set *A ⊂ R is a.e.-open if and only if it is d-open and |A| = |int A|.*

*For each set T ⊂ R and a, b ∈ R, a < b, we denote by ϕ* _{T} *(a, b) the* *measure of the greatest interval J contained in (a, b) \ T if any such interval* *exists, and 0 otherwise. We say that a set T ⊂ R is porous at x ∈ R from* the left if

_{T}

### lim sup

*η→0*

^{+}

*ϕ* _{T} *(x − η, x)* *η* *> 0.*

_{T}

### Being porous from the right is defined similarly.

*We say that a set T ⊂ R is non-degenerate at x ∈ R from the left if* lim sup

*η→0*

^{+}

*|T ∩ (x − η, x)|*

*η* *> 0.*

### Non-degeneracy from the right is defined similarly.

*Let Y be a topological space and let f : R → Y . We say that*

*• f is a Darboux function if the f -image of each interval is connected,*

*• f is peripherally continuous if*

*f* ^{−1} *(U ) ∩ (x − η, x) 6= ∅ 6= f* ^{−1} *(U ) ∩ (x, x + η)* *for each x ∈ R, each neighborhood U ⊂ Y of f (x) and each η > 0.*

^{−1}

^{−1}

*Given two functions f, g : R → R, we define their diagonal product*

*f 4g : R → R* ^{2} *by (f 4g) (x) = (f (x), g(x)).*

*Let A ⊂ B* 1 *. The maximal diagonal class of A with respect to connectivity* *is the family M* _{4} *(A) of those Baire one functions whose diagonal product* *with each function from A has a connected graph.*

_{4}

**III. Auxiliary lemmas. There are many conditions which are equiva-** *lent to the connectivity of the graph of a function f ∈ B* _{1} (cf. [1, Theorem 1.1, p. 9]). For our purpose we will generalize two of them.

*Lemma 1. Assume that Y is a T* 1 *-space and f : R → Y is a Darboux* *function. Then f is peripherally continuous.*

*P r o o f. Suppose that f is not peripherally continuous from the left at* *some x ∈ R. Let a neighborhood U ⊂ Y of f (x) and η > 0 be such that* *f* ^{−1} *(U ) ∩ (x − η, x) = ∅. Set A = f ((x − η, x]) and V = Y \ {f (x)}. Since Y* *is a T* _{1} *-space, V is open. Hence A* _{1} *= A ∩ U = {f (x)} and A* _{2} *= A ∩ V are* *open in A, disjoint and non-empty, and A* 1 *∪ A* 2 *= A. So A is not connected* *and f is not a Darboux function, which completes the proof.*

^{−1}

*Lemma 2. Assume that Y is a metric space and f ∈ B* _{1} *(R, Y ) is periph-* *erally continuous. Then the graph of f is connected.*

*P r o o f. Suppose that Γ (f ) is not connected. Then there exist disjoint* *non-empty sets E* 1 *, E* 2 *⊂ Γ (f ), open in Γ (f ), such that E* 1 *∪ E* 2 *= Γ (f ).*

*For j ∈ {1, 2} let D* _{j} *be the preimage of E* _{j} *under the map x 7→ (x, f (x)).*

_{j}

_{j}

*Then D* _{1} *, D* _{2} *are disjoint, non-empty and D* _{1} *∪ D* _{2} *= R, so fr D* _{1} *6= ∅.*

*By [4] (§31.X.5, p. 397), the set of points of continuity of f | fr D* 1 is non- *empty. Let x be one. By symmetry, we may assume that x ∈ D* _{1} and that *(x − τ, x) ∩ D* _{2} *6= ∅ for each τ > 0. Since E* _{1} *and E* _{2} are disjoint and open *in Γ (f ), there exists a neighborhood U ⊂ Y of f (x) and a τ* 1 *> 0 such that* *f (t) 6∈ U for t ∈ (x − τ* _{1} *, x + τ* _{1} *) ∩ D* _{2} *. Since f | fr D* _{1} *is continuous at x, there* *exists a τ* _{2} *∈ (0, τ* _{1} *) such that f (t) ∈ U for each t ∈ (x − τ* _{2} *, x] ∩ fr D* _{1} . So *D* 2 *∩ (x − τ* 2 *, x) ∩ fr D* 1 *= ∅, whence D* 2 *∩ (x − τ* 2 *, x) 6= ∅ is open. Let (a, b) be* *a component of D* _{2} *∩ (x − τ* _{2} *, x). Then b ∈ (x − τ* _{2} *, x] ∩ fr D* _{1} *and f (b) ∈ U .* *But f (t) 6∈ U for t ∈ (a, b). So f is not peripherally continuous at b from the* left, contrary to the assumption.

### The next lemma follows easily from the definitions.

*Lemma 3. Assume that Y is an arbitrary topological space and that the* *graph of f : R → Y is connected. Then f is a Darboux function.*

### From the above three lemmas we get the following theorem.

*Theorem 4. Let Y be a metric space and f ∈ B* 1 *(R, Y ). Then the fol-* *lowing conditions are equivalent:*

*(A) f is a Darboux function,*

*(B) f is peripherally continuous,* *(C) f has a connected graph.*

### We will also need the following lemma.

*Lemma 5 [6, Lemma 9]. Assume that {J* *n* *: n ∈ N} is a family of non-* *overlapping intervals and (r* _{n} *) is a sequence of non-negative numbers with* P _{∞}

_{n}

_{∞}

*n=1* *r* _{n} *< ∞. Then there exists a sequence (t* _{n} *) such that |t* _{n} *| = 1 for each* *n ∈ N and for every interval I ⊂ R,*

_{n}

_{n}

_{n}

### X

*J*

_{n}*⊂I*

*t* _{n} *r* _{n}

_{n}

_{n}

* ≤ 2 sup {r* *n* *: J* _{n} *⊂ I}.*

_{n}

**IV. Main results. First we will deal with the family C** _{ap} .

**IV. Main results. First we will deal with the family C**

*Proposition 6. Let f ∈ B* 1 *. Then the following conditions are equiva-* *lent:*

*(i) f ∈ M* _{4} *(C* _{ap} ), *(ii) f ∈ M* _{4} *(bC* _{ap} ),

_{4}

_{4}

*(iii) for each x ∈ R and each ε > 0 the set f* ^{−1} *([f (x) − ε, f (x) + ε]) is* *bilaterally non-degenerate at x.*

^{−1}

*P r o o f. The implication (i)⇒(ii) is obvious.*

*(ii)⇒(iii). Suppose that for some x ∈ R and some ε > 0, C is not non-* *degenerate at x from the left, where C = f* ^{−1} *([f (x) − ε, f (x) + ε]). For each* *n ∈ N set x* _{n} *= x − 1/n, I* _{n} *= [x* _{n} *, x* _{n+1} *] and A* _{n} *= I* _{n} *\ C, find a closed set* *B* _{n} *⊂ d-int A* _{n} *such that |A* _{n} *\ B* _{n} *| ≤ |A* _{n} *|/n, and use Lemma 12 of [10] to* *find an approximately continuous function g* *n* *such that 0 ≤ g* *n* *≤ 1 on R,* *g* _{n} *= 1 off A* _{n} *and g* _{n} *= 0 on B* _{n} . Define

^{−1}

_{n}

_{n}

_{n}

_{n+1}

_{n}

_{n}

_{n}

_{n}

_{n}

_{n}

_{n}

_{n}

_{n}

_{n}

_{n}

*g(t) =*

### 0 *if t ≤ x* 1 *or t ≥ x,* *εg* _{n} *(t) if t ∈ I* _{n} *, n ∈ N.*

_{n}

_{n}

*Then clearly g is bounded and approximately continuous except possibly at* *x, and it is continuous at x from the right. Moreover, for each t ∈ [x* _{n} *, x* _{n+1} ],

_{n}

_{n+1}

*|{z ∈ [t, x] : g(z) − g(x) = 0}|*

*x − t* *≥* 1

*x − x* _{n}

_{n}

### [ *∞* *k=n+1*

*B* *k*

*≥* *n*

*n + 1*

*1 −* 1 *n + 1*

### *|[x* *n+1* *, x] \ C|*

*x − x* _{n+1} *.* *So g is approximately continuous at x from the left. But for each t ∈ [x* _{1} *, x)* *either t 6∈* S _{∞}

_{n+1}

_{∞}

*n=1* *A* *n* *, so that g(t) = g(t) − g(x) = ε, or t ∈* S _{∞}

_{∞}

*n=1* *A* *n* , and *then t 6∈ C and |f (t) − f (x)| > ε. Hence f 4g is not peripherally continuous* *at x from the left and by Theorem 4, f 6∈ M* _{4} *(bC* _{ap} ).

_{4}

*Similarly we proceed if C is not non-degenerate at x from the right.*

*(iii)⇒(i). Let g ∈ C* ap *. Fix an x ∈ R and an ε > 0. Let C be as above* *and D = g* ^{−1} *([g(x) − ε, g(x) + ε]). Since C is non-degenerate at x from the* *left, we can find a sequence a* _{n} *% x such that*

^{−1}

_{n}

*n→∞* lim

*|[a* *n* *, x] ∩ C|*

*x − a* _{n} *= 2c > 0.*

_{n}

*Let η ∈ (0, ε) be such that*

*|[a* *n* *, x] ∩ C|*

*x − a* _{n} *> c* *if a* *n* *> x − η, n ∈ N,* and

_{n}

*|[t, x] ∩ D|*

*x − t* *> 1 − c* *if t ∈ (x − η, x).*

*Then |[a* _{n} *, x) ∩ C ∩ D| > 0 for each n ∈ N with a* _{n} *> x − η. Hence f 4g is* *peripherally continuous at x from the left.*

_{n}

_{n}

*Similarly we can prove that f 4g is peripherally continuous at x from the* *right. This implies that the graph of f 4g is connected, so f ∈ M* *4* *(C* ap ).

*Now we turn to the family D.*

*Proposition 7. Let f ∈ B* _{1} *. Then the following conditions are equiva-* *lent:*

*(i) f ∈ M* _{4} *(D),* *(ii) f ∈ M* *4* *(bDC* 0 ),

_{4}

*(iii) for each x ∈ R and each ε > 0 the complement of f* ^{−1} *([f (x) − ε,* *f (x) + ε]) is bilaterally porous at x.*

^{−1}

*P r o o f. The implication (i)⇒(ii) is obvious.*

*(ii)⇒(iii). Suppose that for some x ∈ R and some ε ∈ (0, 1), R \ C is not* *porous at x from the left, where C = f* ^{−1} *([f (x) − ε, f (x) + ε]). Then there* *exists a sequence x* *n* *% x such that x* *n* *6∈ C for each n ∈ N and*

^{−1}

### (1) lim

*n→∞*

*x* *n+1* *− x* *n*

*x − x* _{n} *= 0.*

_{n}

*For each n ∈ N set A* _{n} *= [x* _{n} *, x* _{n+1} *] \ C. We consider two cases:*

_{n}

_{n}

_{n+1}

*• Assume that int A* *n* *6= ∅ for each n ∈ N. Let y* *n* *∈ (a* *n* *, b* *n* *) ⊂ int A* *n* . *For each n ∈ N put J* _{n} *= [y* _{n} *, y* _{n+1} *], define a continuous function g* _{n} by

_{n}

_{n}

_{n+1}

_{n}

*g* _{n} *(t) =*

_{n}

###

###

### 0 *if t ≤ y* _{n} *or t ≥ y* _{n+1} *,* 1 *if t ∈ [b* *n* *, a* *n+1* *],*

_{n}

_{n+1}

### linear *in [y* _{n} *, b* _{n} *] and [a* _{n+1} *, y* _{n+1} ], *and set r* _{n} = R

_{n}

_{n}

_{n+1}

_{n+1}

_{n}

*J*

*n*

*g* _{n} *. Choose a sequence (t* _{n} ) according to Lemma 5. Define *g(t) =*

_{n}

_{n}

### 0 *if t ≤ a* _{1} *or t ≥ x,*

*t* *n* *εg* *n* *(t) if t ∈ J* *n* *, n ∈ N.*

*Then clearly g is bounded and continuous except possibly at x, and it is* *continuous at x from the right. Moreover, for each t ∈ J* _{n} ,

_{n}

### 1 *x − t*

## R *x* *t*

*g* * ≤* 1

*x − t*

### X *∞* *k=n+1*

## R

*J*

_{k}*g* + R

*J*

*n*

*|g|*

*≤* *3ε sup{|J* _{k} *| : k ≥ n}*

_{k}

*x − t*

*≤ 3ε sup*

### *x* _{k+2} *− x* _{k}

_{k+2}

_{k}

*x − x* _{n+2} *: k ≥ n*

_{n+2}

*≤ 3ε sup*

### *x* _{k+2} *− x* _{k} *x − x* _{k} *·*

_{k+2}

_{k}

_{k}

*1 −* *x* _{k+2} *− x* _{k} *x − x* _{k}

_{k+2}

_{k}

_{k}

### _{−1}

_{−1}

*: k ≥ n*

### *.* *So by (1), g is a derivative, whence g ∈ bDC* 0 *. But for each t ∈ [a* 1 *, x) either* *t 6∈* S _{∞}

_{∞}

*n=1* *(a* _{n} *, b* _{n} *), so that |g(t)| = |g(t) − g(x)| = ε, or t ∈* S _{∞}

_{n}

_{n}

_{∞}

*n=1* *(a* _{n} *, b* _{n} ), *in which case t 6∈ C and |f (t) − f (x)| > ε. It follows that f 4g is not* *peripherally continuous at x from the left and f 6∈ M* *4* *(bDC* 0 ).

_{n}

_{n}

*• Assume that int A* _{n} *= ∅ for some n ∈ N. Then C is residual in* *[x* _{n} *, x* _{n+1} *]. Define ε* ^{0} *= (|f (x* _{n} *) − f (x)| − ε)/2 and C* ^{0} *= f* ^{−1} *([f (x* _{n} *) − ε* ^{0} *,* *f (x* *n* *) + ε* ^{0} *]). Then C* ^{0} *is a G* *δ* *set and C* ^{0} *∩ C = ∅, so C* ^{0} is nowhere dense in *[x* _{n} *, x* _{n+1} *]. Hence the complement of C* ^{0} *is not porous at x* _{n} from the right *and for every interval J, int(J \ C* ^{0} *) 6= ∅. Now proceed as in the previous* *case, using x* *n* *instead of x and ε* ^{0} *instead of ε.*

_{n}

_{n}

_{n+1}

^{0}

_{n}

^{0}

^{−1}

_{n}

^{0}

^{0}

^{0}

^{0}

^{0}

_{n}

_{n+1}

^{0}

_{n}

^{0}

^{0}

*Similarly we proceed if R \ C is not porous at x from the right.*

*(iii)⇒(i). Let g ∈ D and let G be its primitive. Fix an x ∈ R and an* *ε > 0. Define C as above. Since R \ C is porous at x from the left, we can* *find a sequence a* *n* *% x such that [a* *2n−1* *, a* *2n* *] ⊂ C and*

*n→∞* lim

*a* _{2n} *− a* _{2n−1}

_{2n}

_{2n−1}

*x − a* _{2n−1} *= 2c > 0.*

_{2n−1}

*Let η ∈ (0, ε) be such that for each t ∈ (x − η, x),* *a* _{2n} *− a* _{2n−1}

_{2n}

_{2n−1}

*x − a* *2n−1*

*> c* *if a* _{2n−1} *> x − η, n ∈ N,*

_{2n−1}

### and

### *G(t) − G(x)*

*t − x* *− g(x)* * ≤* *cε*

### 2 *if t ∈ (x − η, x).*

*Then for each n ∈ N with a* *2n−1* *> x − η, we get*

### *G(a* *2n* *) − G(a* *2n−1* )

*a* _{2n} *− a* _{2n−1} *− g(x)*

_{2n}

_{2n−1}

### * ≤* *2x − a* *2n* *− a* *2n−1*

*a* _{2n} *− a* _{2n−1} *·* *cε* 2 *< ε.*

_{2n}

_{2n−1}

*Hence there exists a t ∈ [a* _{2n−1} *, a* _{2n} *] such that |g(t) − g(x)| ≤ ε. But since*

_{2n−1}

_{2n}

*[a* *2n−1* *, a* *2n* *] ⊂ C, also |f (t) − f (x)| ≤ ε. It follows that f 4g is peripherally*

*continuous at x from the left.*

*Similarly we can prove that f 4g is peripherally continuous at x from the* *right. This implies that the graph of f 4g is connected, so f ∈ M* _{4} *(D).*

_{4}

*R e m a r k. By Proposition 7, we get M* *4* *(DC* *a.e.* *) = M* *4* *(D). It turns* *out that the analogous result does not hold for the family C* _{ap} .

*Proposition 8. Let f ∈ B* _{1} *. Then the following conditions are equiva-* *lent:*

*(i) f ∈ M* _{4} *(C* _{ap} *C* _{a.e.} ), *(ii) f ∈ M* _{4} *(bC* _{ap} *C* _{0} ),

_{4}

_{a.e.}

_{4}

*(iii) for each x ∈ R and each ε > 0 the closure of f* ^{−1} *([f (x)− ε, f (x)+ ε])* *is bilaterally non-degenerate at x.*

^{−1}

*P r o o f. The implication (i)⇒(ii) is obvious.*

*(ii)⇒(iii). Suppose that for some x ∈ R and some ε > 0, cl C is not* *non-degenerate at x from the left, where C = f* ^{−1} *([f (x) − ε, f (x) + ε]). For* *each n ∈ N set x* _{n} *= x − 1/n, I* _{n} *= [x* _{n} *, x* _{n+1} *] and A* _{n} *= int I* _{n} *\ cl C, find* *a closed set B* *n* *⊂ A* *n* *such that |A* *n* *\ B* *n* *| ≤ |A* *n* *|/n, and find a continuous* *function g* _{n} *such that 0 ≤ g* _{n} *≤ 1 on R, g* _{n} *= 1 off A* _{n} *and g* _{n} *= 0 on B* _{n} . Define

^{−1}

_{n}

_{n}

_{n}

_{n+1}

_{n}

_{n}

_{n}

_{n}

_{n}

_{n}

_{n}

_{n}

*g(t) =*

### 0 *if t ≤ x* 1 *or t ≥ x,* *εg* _{n} *(t) if t ∈ I* _{n} *, n ∈ N.*

_{n}

_{n}

*Then clearly g ∈ bC* _{0} . Repeating the argument of Proposition 6 one can see *that g ∈ C* ap *and f 4g is not peripherally continuous at x from the left.*

*Hence f 6∈ M* _{4} *(bC* _{ap} *C* _{a.e.} ).

_{4}

_{a.e.}

*Similarly we proceed if cl C is not non-degenerate at x from the right.*

*(iii)⇒(i). Let g ∈ C* ap *C* *a.e.* *. Fix an x ∈ R and an ε > 0. Let C be as above* *and D = g* ^{−1} *((g(x) − ε, g(x) + ε)). Since cl C is bilaterally non-degenerate* *at x, we can find a sequence a* _{n} *% x such that*

^{−1}

_{n}

*n→∞* lim

*|[a* _{n} *, x] ∩ cl C|*

_{n}

*x − a* _{n} *= 2c > 0.*

_{n}

*Let η ∈ (0, ε) be such that*

*|[a* _{n} *, x] ∩ cl C|*

_{n}

*x − a* *n* *> c* *if a* *n* *> x − η, n ∈ N,* and

*|[t, x] ∩ D|*

*x − t* *> 1 − c* *if t ∈ (x − η, x).*

*Then for each n ∈ N with a* *n* *> x − η, we get*

*|(a* _{n} *, x) ∩ cl C ∩ D| = |(a* _{n} *, x) ∩ cl C ∩ int D| > 0.*

_{n}

_{n}

*(We used the fact that D is a.e.-open.) Hence (a* *n* *, x) ∩ C ∩ D 6= ∅, so f 4g*

*is peripherally continuous at x from the left.*

*Similarly we prove that f 4g is peripherally continuous at x from the* *right. This implies that the graph of f 4g is connected and that f ∈* *M* _{4} *(C* _{ap} *C* _{a.e.} ).

_{4}

_{a.e.}

*Proposition 9. M* _{4} *(C* _{ap} *C* _{a.e.} *) \ M* _{4} *(C* _{ap} *) 6= ∅.*

_{4}

_{a.e.}

_{4}

*P r o o f. For each n ∈ N set a* _{n} = 1 and find a nowhere dense perfect *set F* _{n} *such that inf F* _{n} *= a* _{n+1} *, sup F* _{n} *= a* _{n} *and |F* _{n} *| = (1−1/n)(a* _{n} *−a* _{n+1} ).

_{n}

_{n}

_{n}

_{n+1}

_{n}

_{n}

_{n}

_{n}

_{n+1}

### Observe that

### (2) *0 ∈ d-int*

*(−∞, 0] ∪* [ *∞* *n=1*

*F* *n*

### *.*

*Fix an n ∈ N. Let {E* _{n,k} *: k ∈ N} be the family of all components of* *(a* *n+1* *, a* *n* *) \ F* *n* *. For all k ∈ N if E* *n,k* *= (b* *n,k* *− c* *n,k* *, b* *n,k* *+ c* *n,k* ), then let *f* _{n,k} *be a continuous function such that f* _{n,k} *(b* _{n,k} *) = 1 and f* _{n,k} *(x) = 0 if*

_{n,k}

_{n,k}

_{n,k}

_{n,k}

_{n,k}

*|x − b* _{n,k} *| ≥ max {c* _{n,k} *, c* ^{2} _{n,k} *}. Define*

_{n,k}

_{n,k}

_{n,k}

*f (x) =*

### ( *f* _{n,k} *(x) if x ∈ E* _{n,k} *, n, k ∈ N,*

_{n,k}

_{n,k}

### 1 *if x ≤ 0,*

### 0 otherwise.

*Then evidently f is approximately continuous on R\{0} and it is continuous* at 0 from the left, so at these points condition (iii) of Proposition 8 is *satisfied. Fix an ε ∈ (0, 1) and set C = f* ^{−1} *([1−ε, 1+ε]). Then C∩* S _{∞}

^{−1}

_{∞}

*n=1* *F* _{n} =

_{n}

*∅ and since C ∩ E* _{n,k} *6= ∅ for all n, k ∈ N, we have cl C ⊃* S _{∞}

_{n,k}

_{∞}

*n=1* *F* _{n} . Hence *by (2), f does not satisfy condition (iii) of Proposition 6 and it does satisfy* condition (iii) of Proposition 8, which completes the proof.

_{n}

### R e m a r k. It is easy to see that the function constructed in the proof of Proposition 9 is discontinuous on a set of positive measure. This leads *to conjecturing that for functions from C* *a.e.* the proposition analogous to Proposition 7 holds. This is indeed true.

*Proposition 10. M* _{4} *(C* _{ap} *C* _{a.e.} *) ∩ C* _{a.e.} *= M* _{4} *(C* _{ap} *) ∩ C* _{a.e.} *.* *P r o o f. The inclusion “⊃” is obviously satisfied.*

_{4}

_{a.e.}

_{a.e.}

_{4}

_{a.e.}

*Let f ∈ M* _{4} *(C* _{ap} *C* _{a.e.} *) ∩ C* _{a.e.} *, x ∈ R and ε > 0. Set C = f* ^{−1} *([f (x) − ε,* *f (x) + ε]). By Proposition 8, cl C is bilaterally non-degenerate at x. Since* *t ∈ C implies |f (t) − f (x)| ≤ ε, it follows that for every t ∈ cl C, if t is* *a point of continuity of f , then |f (t) − f (x)| ≤ ε. Hence |cl C \ C| = 0* *and C is bilaterally non-degenerate at x, i.e., condition (iii) of Proposition 6* is satisfied.

_{4}

_{a.e.}

_{a.e.}

^{−1}

**References**

### [1] *A. M. B r u c k n e r, Differentiation of Real Functions, Lecture Notes in Math. 659,*

### Springer, Berlin, 1978.

### [2] *A. M. B r u c k n e r, J. M aˇr´ık and C. E. W e i l, Baire one, null functions, in: Con-* temp. Math. 42, Amer. Math. Soc., 1985, 29–41.

### [3] *C. G o f f m a n, C. J. N e u g e b a u e r and T. N i s h i u r a, Density topology and approx-* *imate continuity, Duke Math. J. 28 (1961), 497–506.*

### [4] *K. K u r a t o w s k i, Topology, Vol. I, Academic Press, New York, 1966.*

### [5] J. L u k eˇs, J. M a l ´ *y and L. Z a j´ıˇce k, Fine Topology Methods in Real Analysis and* *Potential Theory, Lecture Notes in Math. 1189, Springer, Berlin, 1986.*

### [6] *A. M a l i s z e w s k i, Characteristic functions and products of bounded derivatives,* Proc. Amer. Math. Soc., to appear.

### [7] *C. J. N e u g e b a u e r, On a paper by M. Iosifescu and S. Marcus, Canad. Math. Bull.*

### 6 (1963), 367–371.

### [8] *R. J. O’ M a l l e y, Approximately continuous functions which are continuous almost* *everywhere, Acta Math. Acad. Sci. Hungar. 33 (1979), 395–402.*

### [9] *G. P e t r u s k a and M. L a c z k o v i c h, Baire 1 functions, approximately continuous* *functions and derivatives, ibid. 25 (1974), 189–212.*

### [10] *Z. Z a h o r s k i, Sur la premi`ere d´eriv´ee, Trans. Amer. Math. Soc. 69 (1950), 1–54.*

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