UDC 517.513

MATHEMATICS

V. A. STARTSEV

ON SYMMETRIC CONTINUITY AND SYMMETRIC DIFFERENTIABILITY WITH RESPECT TO SETS

(Presented by Academician P. S. Novikov on 13 IX 1968)

1. Let \(f(x)\) be a real function defined on the interval \((0,1)\); let \(k\) be an arbitrary fixed natural number;

\[ \Delta^k f(x,h)= \sum_{i=0}^{k}(-1)^i {k \choose i} f[x+(k-2i)h] \]

is the symmetric difference of order \(k\) at the point \(x\); and let \(Q\) be a set of points \(h\) on the line having zero as a limit point and situated to the right of zero.

We shall call \(f(x)\) \(Q\)-symmetrically continuous of order \(k\) at the point \(x\in(0,1)\) if

\[ \lim_{h\to 0,\ h\in Q}\Delta^k f(x,h)=0. \]

We shall call \(f(x)\) \(Q\)-symmetrically differentiable of order \(k\) at the point \(x\in(0,1)\) if there exists

\[ D_Q^k f(x)=\lim_{h\to 0,\ h\in Q}\frac{\Delta^k f(x,h)}{(2h)^k}. \]

The upper and lower \(Q\)-symmetric derivative numbers are defined as follows:

\[ \overline{D}_Q^k f(x)= \overline{\lim}_{h\to 0,\ h\in Q}\frac{\Delta^k f(x,h)}{(2h)^k}, \qquad \underline{D}_Q^k f(x)= \underline{\lim}_{h\to 0,\ h\in Q}\frac{\Delta^k f(x,h)}{(2h)^k}. \]

If zero is an interior point for the set \(Q\), then \(Q\)-symmetric continuity of order \(k\) coincides with the usual symmetric continuity of order \(k\), while the definition of the \(Q\)-symmetric derivative coincides with the definition of the Riemann derivative of order \(k\). Recall that a function \(f(x)\) has a Riemann derivative of order \(k\) at the point \(x\) if there exists

\[ D^k f(x)=\lim_{h\to 0}\frac{\Delta^k f(x,h)}{(2h)^k}. \]

For \(k=1\) this definition coincides with the definition of the symmetric derivative; for \(k=2\), with the definition of the Schwarz derivative.

It is clear that ordinary continuity at a point \(x\) implies \(Q\)-symmetric continuity of any order \(k\), and ordinary differentiability implies \(Q\)-symmetric differentiability of order \(1\) at this point \(x\), whatever the set \(Q\) may be. The converse (for a fixed set \(Q\)) does not hold. However, as follows from the theorems given below, if one assumes that \(Q\)-symmetric continuity or \(Q\)-symmetric differentiability of order \(k\) of the function \(f(x)\) holds at every point of a set \(E\) of positive measure, then, under certain conditions imposed on \(Q\), it follows that \(f(x)\) is ordinarily continuous or ordinarily differentiable almost everywhere on \(E\).

We shall assume that for all \(x\) the set \(Q\) is one and the same. The idea of such a generalization is due to G. Kh. Sindalovskii. He introduced \((^1)\) the concepts of continuity and differentiability with respect to congruent sets (the concepts of \(Q\)-continuity and \(Q\)-differentiability) and studied the connection of these concepts with the concepts of ordinary continuity and differentiability.

Let us introduce the following classes of sets \(Q\) having zero as their limit point.

To class (A) we assign those sets \(Q\) that have positive measure in every neighborhood of zero, i.e. such that \(\operatorname{mes} Q \cap (0,\delta) > 0\) for every \(\delta > 0\).

To class (B) we assign those sets \(Q\) that have positive lower density at zero, i.e. those for which
\[ \lim_{\delta\to 0}\frac{\operatorname{mes} Q\cap(0,\delta)}{\delta}>0. \]

To class (C) we assign those sets \(Q\) whose closure \(\overline Q\) belongs to class (B).

Theorem 1. Let \(k\) be an arbitrary fixed natural number, \(f(x)\) a function measurable on \((0,1)\), for which
\[ \lim_{\substack{h\to 0\\ h\in Q}}\Delta^k f(x,h)=0 \]
at every point of a set \(E,\ E\subset(0,1),\ \operatorname{mes} E>0\), and let the set \(Q\) belong to class (A). Then \(f(x)\) is continuous almost everywhere on \(E\).

For every set \(Q\) not belonging to class (A), there exists an everywhere on \((0,1)\) symmetrically discontinuous* (and hence everywhere discontinuous in the ordinary sense) measurable function that will be \(Q\)-symmetrically continuous of order \(k\) almost everywhere on \((0,1)\).

Theorem 2. Let \(k\) be an arbitrary fixed natural number, and let the set \(Q\) belong to class (B). If \(f(x)\) is a function measurable on \((0,1)\) for which
\[ \lim_{\substack{h\to 0\\ h\in Q}}\left|\Delta^k f(x,h)/(2h)^k\right|<+\infty \]
and \(f'_{ac}(x)\) exists at every point \(x\in E,\ E\subset(0,1),\ \operatorname{mes} E>0\), then \(f(x)\) is differentiable in the ordinary sense almost everywhere on \(E\).

For every set \(Q\) not belonging to class (B), there exists a function measurable on \((0,1)\) that has a \(Q\)-symmetric derivative of order \(k\) almost everywhere on \((0,1)\) and nowhere has a symmetric (and hence ordinary) derivative.

Theorem 3. Let \(k\) be an arbitrary fixed natural number, and let the set \(Q\) belong to class (C). If \(f(x)\) is a function continuous on \((0,1)\) for which
\[ \lim_{\substack{h\to 0\\ h\in Q}}\left|\frac{\Delta^k f(x,h)}{(2h)^k}\right|<+\infty \]
and \(f'_{ac}(x)\) exists at every point \(x\in E,\ E\subset(0,1),\ \operatorname{mes} E>0\), then \(f(x)\) is differentiable in the ordinary sense almost everywhere on \(E\).

For every set \(Q\) not belonging to class (C), one can construct a function continuous on \((0,1)\) having a finite \(Q\)-symmetric derivative of order \(k\) on some set \(E,\ E\subset(0,1)\), and having no ordinary derivative on it. The measure of the set \(E\) may be arbitrarily close to one.

For \(k=1\) the following assertions are valid:

Theorem 4. Let \(f(x)\) be a function measurable on \((0,1)\); \(Q\) a set belonging to class (B). Then \(\overline D_Q f(x)\) almost everywhere on \((0,1)\) is equal to \(\overline D f(x)\).

For every set \(Q\) not belonging to class (B), there exists a function \(\varphi(x)\), measurable on \((0,1)\) and nowhere symmetrically differentiable, for which almost everywhere on \((0,1)\) the relation
\[ \overline D_Q\varphi(x)<\overline D\varphi(x)=+\infty \]
holds (here \(\overline D f(x)\) is the upper symmetric derivative number of the function \(f(x)\)).

Theorem 5. Let \(f(x)\) be a function continuous on \((0,1)\); \(Q\) a set—

* A function \(f(x)\) is called everywhere symmetrically discontinuous on \((0,1)\) if for every \(x\in(0,1)\) \(\Delta^1 f(x,h)\) does not tend to zero as \(h\to 0\).

a set belonging to class (C). Then almost everywhere on \((0,1)\)
\[ \overline D_Q f(x)=\overline D f(x). \]

For every set \(Q\) not belonging to class (C), there exists a continuous function \(\varphi(x)\), nowhere differentiable on \((0,1)\), for which almost everywhere the relation
\[ \overline D_Q\varphi(x)<\overline D\varphi(x)=+\infty \]
holds.

Analogous assertions hold for the lower symmetric derived numbers.

Theorem 6. Let \(f(x)\) be a measurable function on \((0,1)\); let \(Q\) be a set belonging to class (B). If at each point \(x\in E\subset(0,1)\) at least one of the inequalities
\[ \overline D_Q f(x)<+\infty \]
or
\[ \underline D_Q f(x)>-\infty \]
holds, then \(f(x)\) is differentiable in the ordinary sense almost everywhere on \(E\).

For every set \(Q\) not belonging to class (B), there exists a measurable symmetric function \(\varphi(x)\), nowhere symmetrically differentiable on \((0,1)\), for which almost everywhere on \((0,1)\) the inequality
\[ \overline D_Q\varphi(x)<+\infty \]
holds.

Theorem 7. Let \(f(x)\) be a continuous function on \((0,1)\); let \(Q\) be a set belonging to class (C). If at each point \(x\in E\subset(0,1)\) at least one of the inequalities
\[ \overline D_Q f(x)<+\infty \]
or
\[ \underline D_Q f(x)>-\infty \]
holds, then \(f(x)\) is differentiable in the ordinary sense almost everywhere on \(E\).

For every set \(Q\) not belonging to class (C), there exists a function \(\varphi(x)\) continuous on \((0,1)\), having no ordinary derivative, but satisfying the inequality
\[ \overline D_Q\varphi(x)<+\infty \]
almost everywhere on \((0,1)\).

2. The proof of Theorem 2 is based on the following assertions:

Lemma 1. Let \(k\) be a fixed natural number; let \(f(x)\) be a function measurable in the sense of Lebesgue on \((0,1)\); let \(Q\) be a set belonging to class (B); and let \(F\) be a Borel-measurable set such that: a) \(F\subset(0,1)\), \(\operatorname{mes}F=1\); b) \(f(x)\) is Borel-measurable on \(F\); c) for every \(x\in CF\) \((CF=(0,1)\setminus F)\) there exists a sequence of numbers \(\{c_i\}\), \(c_i\in F\), such that \(c_i\to x\) and \(f(c_i)\to f(x)\) as \(i\to\infty\). If
\[ \overline D_Q^{\,k}f(x)<+\infty \]
for \(x\in E\), \(E\subset(0,1)\), \(\operatorname{mes}E>0\), then there exist a perfect set \(P\), \(P\subset E\cap F\), of measure arbitrarily close to the measure of \(E\), and numbers \(\eta_0>0\), \(l_0>0\), such that
\[ \Delta^k f(x,h)/(2h)^k<\eta_0 \]
for \(x\in P\), \(h\in Q\), \(h<l_0\), such that \(x\pm rh\in F\) \((1\le r\le k)\).

Lemma 1 is proved by the methods developed by G. Kh. Sindalovskii \((^{1,2})\). The first part of Theorem 3 follows from the first part of Theorem 2. Theorem 4 is based on the following assertion:

Lemma 2. Let \(f(x)\) be a measurable function on \((0,1)\); let \(Q\) be a set belonging to class (B). If
\[ \overline D_Q f(x)<+\infty \]
on some set \(E\) of positive measure, then almost everywhere on \(E\) \(f(x)\) is asymptotically differentiable. An analogous assertion holds if
\[ \underline D_Q f(x)>-\infty. \]

Theorems 5 and 6 follow from Theorem 4; Theorem 7 follows from Theorem 6.

Moscow State
Pedagogical Institute named after V. I. Lenin

Received
9 IX 1968

CITED LITERATURE

1. G. Kh. Sindalovskii, Izv. AN SSSR, Ser. Matem., 26, 125 (1962).
2. G. Kh. Sindalovskii, Izv. AN SSSR, Ser. Matem., 24, 707 (1960).