Mathematics

G. Kh. Sindalovskii

On Continuity and Differentiability with Respect to Congruent Sets

(Presented by Academician P. S. Aleksandrov, 27 V 1960)

Let \(f(x)\) be a real function, taking finite values, defined and measurable on the interval \((a,b)\). Fix some measurable set \(Q\) in a neighborhood of zero, having zero as a limit point.

We define the notion of \(Q\)-continuity of a function. A function \(f(x)\) is called \(Q\)-continuous at a point \(x \in (a,b)\) if

\[ \lim_{h\to 0,\ h\in Q} [f(x+h)-f(x)] = 0 \]

(i.e., the limit is considered under the condition that \(h \to 0\) while remaining in the set \(Q\)).

Analogously one defines the \(Q\)-derivative of the function \(f(x)\) at the point \(x\):

\[ f'_Q(x)=\lim_{h\to 0,\ h\in Q} \frac{f(x+h)-f(x)}{h}. \]

If the set \(Q\) is an interval containing zero, then \(Q\)-continuity and \(Q\)-differentiability are, respectively, ordinary continuity and differentiability.

It is clear that from the ordinary continuity or differentiability of a function at some point \(x\) there follows, respectively, \(Q\)-continuity or \(Q\)-differentiability at this point, whatever \(Q\) may be; the converse (for fixed \(Q\)) need not hold.

However, if it is assumed that \(Q\)-continuity or \(Q\)-differentiability of the function \(f(x)\) holds at every point of a set \(E \subset (a,b)\) of positive measure, then, under certain conditions imposed on \(Q\), it will follow from this that \(f(x)\) is ordinarily continuous or differentiable almost everywhere on \(E\).

Here it is assumed that for all \(x \in E\) the set \(Q\) is one and the same, i.e., continuity or differentiability is considered with respect to congruent sets.

We find a necessary and sufficient condition that must be imposed on the set \(Q\) (separately for continuity and differentiability) in order that the stated assertion be true.*

We introduce the following three classes of sets \(Q\) having zero as a limit point:

To class (A) we assign those sets \(Q\) which have positive measure in every neighborhood of zero, i.e.

\[ \operatorname{mes} Q\cdot(-\delta,\delta)>0 \]

for every \(\delta>0\).

* Some sufficient conditions (under strong restrictions imposed on \(Q\)) were obtained by us as a by-product in the paper \((^{1})\) (pp. 399, 411, 428).

In class (B) we place those sets \(Q\) that have positive lower density at zero, i.e.

\[ \lim_{\delta\to 0+}\frac{\operatorname{mes} Q\cdot(-\delta,\delta)}{2\delta}>0^{*}. \]

In class (C) we place those sets \(Q\) for which the closure \(\overline Q\) has positive lower density at zero, i.e.

\[ \lim_{\delta\to 0+}\frac{\operatorname{mes}\overline Q\cdot(-\delta,\delta)}{2\delta}>0. \]

Theorem 1. If a function \(f(x)\), measurable on \((a,b)\), is \(Q\)-continuous at every point of a set \(E\) of positive measure, and the set \(Q\) belongs to class (A), then \(f(x)\) is continuous in the ordinary sense almost everywhere on the set \(E\).

For any set \(Q\) not belonging to class (A), one can construct a function, measurable on the interval, that will be \(Q\)-continuous on some set \(E\) of positive measure (and even almost everywhere) and at the same time discontinuous everywhere.

Theorem 2. If a function \(f(x)\), measurable on \((a,b)\), has a finite \(Q\)-derivative \(f'_Q(x)\) at every point of a set \(E\) of positive measure, and the set \(Q\) belongs to class (B), then \(f(x)\) has an ordinary derivative almost everywhere on the set \(E\).

For any set \(Q\) not belonging to class (B), one can construct a function, measurable on the interval, that has a finite \(Q\)-derivative and has no ordinary derivative on some set \(\mathcal E\) of positive measure.

Theorem 2 can be strengthened by replacing the requirement of \(Q\)-differentiability by the requirement of finiteness of the \(Q\)-derivative numbers

\[ \overline f'_Q(x)=\overline{\lim}_{h\to 0,\ h\in Q}\frac{f(x+h)-f(x)}{h},\qquad \underline f'_Q(x)=\underline{\lim}_{h\to 0,\ h\in Q}\frac{f(x+h)-f(x)}{h}. \]

Namely, the following is true.

Theorem 3. A necessary and sufficient condition for the finiteness of the \(Q\)-derivative numbers of a measurable function \(f(x)\) on a set \(E\) of positive measure to imply ordinary differentiability almost everywhere on \(E\) is that the set \(Q\) belong to class (B).

If one considers only functions continuous on \((a,b)\), then the requirements imposed on the set \(Q\) can be weakened.

Theorem 4. If a function \(f(x)\), continuous on \((a,b)\), has finite \(Q\)-derivative numbers at every point of a set \(E\) of positive measure, and the set \(Q\) belongs to class (C), then \(f(x)\) has an ordinary derivative almost everywhere on \(E\).

For any set \(Q\) not belonging to class (C), one can construct a continuous function that has a finite \(Q\)-derivative and has no ordinary derivative on some set \(\mathcal E\) of positive measure.

Received
25 V 1960

CITED LITERATURE

1. G. Kh. Sindalovskii, Izv. AN SSSR, ser. matem., 22, No. 3, 395 (1958).

* Here it is possible that

\[ \lim_{\delta\to 0+}\frac{\operatorname{mes} Q\cdot(0,\delta)}{\delta}=0 \quad\text{and}\quad \lim_{\delta\to 0+}\frac{\operatorname{mes} Q\cdot(-\delta,0)}{\delta}=0. \]