V. M. Tsodyks

ON SETS OF POINTS WHERE THE DERIVATIVE IS RESPECTIVELY FINITE AND INFINITE

(Presented by Academician S. L. Sobolev, 15 I 1957)

In the works of N. N. Luzin (¹), Z. S. Zagorskii (²), and A. L. Brudno (³) it was shown that the set of points at which an infinite derivative exists is an \(F_{\sigma\delta}\) set of measure zero, while the set of points at which at least one derived number of a function of a real variable tends to infinity is a \(G_\delta\) set.

E. M. Landis (⁴), for an arbitrary set \(E\) of type \(F_{\sigma\delta}\) and of measure zero, constructed a continuous function \(F(x)\) such that \(F'(x)=+\infty\) for \(x\in E\), \(\underline F(x)<+\infty\) for \(x\notin E\) (by \(\underline F\) is denoted the lower derived number of the function \(F\)).

In the present note we briefly state a theorem giving a partial answer to the question of the relation between the set of points where the derivative is equal to infinity and the set of points where the derivative is finite (²).

Theorem. Let \(E\) be a set of type \(F_{\sigma\delta}\) of measure zero and \(N\) a set of type \(G_\delta\), lying on the axis \(OX\), with \(N\supset E\). Then there exists a continuous increasing function \(F(x)\) such that \(F'(x)=+\infty\) for \(x\in E\), \(\underline F(x)<+\infty\) for \(x\notin E\), and for \(x\in CN\), \(F'(x)\) exists and is finite.

Proof.

1. Let
   \[ E=\prod_{n=1}^{\infty} E_n \quad\text{and}\quad E_n=\sum_{k=1}^{\infty} E_{nk}, \]
   where the \(E_{nk}\) are closed sets.

Let
\[ N=\prod_{n=1}^{\infty} G_n, \]
where the \(G_n\) are open sets such that \(G_n\supset E\).

We may assume that
\[ \operatorname{mes} G_n<\frac1{2^n},\qquad G_n\supset G_{n+1},\qquad E_n\subset G_n,\qquad E_{nk}\subset E_{n,k+1}, \]
\[ E_{n+1,k}\subset E_{nk}. \]

Let a summable function
\[ u(x)=\sum_{n=1}^{\infty} u_n(x) \]
be given, where \(u_n(x)=1\) for \(x\in G_n\), and \(u_n(x)=0\) for \(x\notin G_n\).

Place in \(E_n\) the set
\[ e_n=\sum_{k=1}^{\infty} e_{nk}, \]
where the \(e_{nk}\) are sets simultaneously of type \(F_\sigma\) and \(G_\delta\), possessing the following properties:

1) \(e_{n,k+1}\supset e_{nk}\), \(E\cdot E_{nk}\subset e_{nk}\subset E_{nk}\).

2) For every integer \(k\ge 2\) there exist six open sets \(g^*_{1k}\), \(g^{*1}_{nk}\), \(g^{*2}_{nk}\), and \(g_{nk}\), \(g^1_{nk}\), \(g^2_{nk}\), and for \(k=1\) three open sets \(g_{n1}\), \(g^1_{n1}\), \(g^2_{n1}\) such that:

a)
\[ G_n=g_{n1}\supset g^1_{n1}\supset g^2_{n1}\supset e_{n1}; \]

b) if \(x_0\in g_{n1}\), then for every \(h\)
\[ \frac{\operatorname{mes}(g^1_{n1})^h}{h}\le \frac1{2^n}, \]
where
\[ (g^1_{n1})^h = g^1_{n1}\cdot [x_0,x_0+h] \]
(or \(g^1_{n1}\cdot [x_0+h,x_0]\), if \(h<0\));

c) \(g_{n,k-1}^{2}-e_{n,k-1}=g_{nk}^{*}\supset g_{nk}^{*1}\supset g_{nk}^{*2}\supset g_{nk}\supset g_{nk}^{1}\supset g_{nk}^{2}\supset (e_{nk}-e_{n,k-1})\), where \(k\geqslant 2\); \(g_{nk}\subset G_k\);

d) the points \(Cg_{nk}\) are points of density for \(Cg_{nk}^{1}\); the points \(Cg_{nk}^{1}\) are points of density for \(Cg_{nk}^{2}\); the points \(Cg_{nk}^{*}\) are points of density for \(Cg_{nk}^{*1}\); the points \(Cg_{nk}^{*1}\) are points of density for \(Cg_{nk}^{*2}\);

e) if \(x_0\notin g_{nk}^{*2}\), then for any \(h\)

\[ \frac{\int_{g_{nk}^{h}} u(\xi)\,d\xi}{h}<\frac{1}{2^k}, \]

where \(g_{nk}^{h}=g_{nk}\cdot [x_0,x_0+h]\) (or \(g_{nk}\cdot [x_0+h,x_0]\), if \(h<0\)).

Obviously, \(E\subset e_n\subset E_n\), \(\prod_{k=1}^{\infty} g_{nk}\subset N\), and the points of the set \(e_n\) are points of density for the set

\[ D_n=\sum_{k=1}^{\infty}\bigl(g_{nk}^{2}-g_{n,k+1}^{*1}\bigr). \]

Let \(e_{n1}=\prod_{l=1}^{\infty} G_{n1}^{(l)}\), where \(G_{n1}^{(l)}\) is an open set, \(G_{n1}^{(l+1)}\subset G_{n1}^{(l)}\subset g_{n1}\), \(\overline{G}_{n1}^{(l+1)}\subset G_{n1}^{(l)}+E_{n1}\).

Let \(e_{nk}-e_{n,k-1}=\prod_{l=1}^{\infty} G_{nk}^{(l)}\) \((k=2,3,\ldots)\), where \(G_{nk}^{(l)}\) is an open set, \(G_{nk}^{(l+1)}\subset G_{nk}^{(l)}\subset g_{nk}\), \(\overline{G}_{nk}^{(l+1)}\subset G_{nk}^{(l)}+E_{nk}\).

Put \(G_n^{(l)}=\sum_{k=1}^{\infty} G_{nk}^{(l)}\) (obviously, \(e_n\subset G_n^{(l)}\subset G_n\)) and construct, for each natural \(l\), open sets \(A_{nl}, A_{nl}^{1}, A_{nl}^{2}\) such that:

a) \(G_{n}^{(1)}\cdot G_1=A_{n1}\supset A_{n1}^{1}\supset A_{n1}^{2}\supset E\);

b) \(A_{n,l-1}^{2}\cdot G_n^{(l)}\cdot G_l=A_{nl}\supset A_{nl}^{1}\supset A_{nl}^{2}\supset E\), where \(l\geqslant 2\);

c) the points \(CA_{nl}\) are points of density for \(CA_{nl}^{1}\); the points \(CA_{nl}^{1}\) are points of density for \(CA_{nl}^{2}\).

1. We now construct auxiliary functions.

For each natural \(n\) we construct asymptotically continuous functions \({}^{(5,6)}\)

1) \(\theta_{nk}(x)\) \((k=1,2,\ldots)\), where \(\theta_{nk}(x)=0\) for \(x\in Cg_{nk}^{1}\); \(\theta_{nk}(x)=1\) for \(x\in g_{nk}^{2}\); \(0\leqslant \theta_{nk}(x)\leqslant 1\) for the remaining points.

2) \(\theta_{n,k+1}^{*}(x)\) \((k=1,2,\ldots)\), where \(\theta_{n,k+1}^{*}(x)=0\) for \(x\in Cg_{n,k+1}^{*1}\); \(\theta_{n,k+1}^{*}(x)=1\) for \(x\in g_{n,k+1}^{*2}\); \(0\leqslant \theta_{n,k+1}^{*}(x)\leqslant 1\) for the remaining points.

3) \(v_{nl}(x)\) \((l=1,2,\ldots)\), where \(v_{nl}(x)=0\) for \(x\in CA_{nl}^{1}\); \(v_{nl}(x)=1\) for \(x\in A_{nl}^{2}\); \(0\leqslant v_{nl}(x)\leqslant 1\) for the remaining points.

Put

\[ \theta_n(x)=\sum_{k=1}^{\infty}\bigl(\theta_{nk}(x)-\theta_{n,k+1}^{*}(x)\bigr);\qquad w_{nl}(x)=\min[\theta_n(x),v_{nl}(x)]. \]

We construct a summable function

\[ w_n(x)=\sum_{l=1}^{\infty} w_{nl}(x). \]

At the points \(CN\) the functions \(\theta_n(x), w_{nl}(x), w_n(x)\) are asymptotically continuous; for \(x\in A_{nl}^{2}\cdot D_n\), \(w_n(x)\geqslant l\).

Let, further,

\[ W_n(x)=\int_0^x w_n(\xi)\,d\xi. \]

Then for \(x\notin E_n\) we have \(0\leq W_n(x)<+\infty\).

1. We proceed to the construction of the desired function. Put
   \[ w_n^*(x)=\min_{m\le n} w_m(x);\qquad w_n^{**}(x)=\max [0, w_n^*(x)-(n-1)]. \]

Let
\[ f_n(x)=\min [1,w_n^{**}(x)]. \]
At points of \(CN\) the functions \(f_n(x)\) are asymptotically continuous; \(f_n(x)\le u_n(x)\); if \(x_0\in E\), then \(x_0\) is a point of density of the set on which \(f_n(x)=1\).

Now put
\[ f(x)=\sum_{n=1}^{\infty} f_n(x),\qquad F(x)=\int_0^x f(\xi)\,d\xi. \]
Observe that
\[ f(x)<w_n^*(x)+n,\qquad F(x)\le W_n(x)+n. \]

The increasing function \(F(x)\), absolutely continuous on every interval of the \(OX\)-axis, satisfies the conditions of the theorem:

I. If \(x_0\in E\), then \(F'(x_0)=+\infty\).

II. If \(x^*\notin E\), then \(\overline{F}(x^*)<+\infty\).

III. If \(x\in CN\), then \(F'(x)\) exists and is finite.

Velikie Luki State
Pedagogical Institute

Received
2 I 1957

CITED LITERATURE

¹ N. N. Luzin, Collected Works, 1, Publishing House of the Academy of Sciences of the USSR, 1953, pp. 5–24. ² Z. S. Zagorskii, Mat. sbornik, 9(51), 3, 487 (1941). ³ A. L. Brudno, Mat. sbornik, 13(55), 1, 119 (1943). ⁴ E. M. Landis, DAN, 107, No. 2 (1956). ⁵ G. P. Tolstov, Mat. sbornik, 8(50), 1, 149 (1940). ⁶ V. S. Bogomolova, Mat. sbornik, 32, 1, 152 (1924).