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}) such that:}), and (g_{nk}), (g^1_{nk}), (g^2_{nk}), and for (k=1) three open sets (g_{n1}), (g^1_{n1}), (g^2_{n1

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).}^{(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).}^{(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)