Approximate differentiation: Jarník points.

*(English)*Zbl 0814.26005If \(f: \mathbb{R}\to \mathbb{R}\) the set \(J_ f\), of Jarník points, consists of all \(x\) such that
\[
ap- \lim_{y\to x} \left| \frac{f(y)- f(x)}{y- x}\right|=\infty.
\]
It is known that there is a continuous \(f\) such that \(J_ f= \mathbb{R}\); in fact, almost every path of one- dimensional Brownian motion gives an example of such a function [D. Geman and J. Horowitz, Ann. Probab. 8, 1-67 (1980; Zbl 0499.60081)]. Also Jarník proved that \(| J_ f|= 0\) for typically continuous functions [V. Jarník, Fundam. Math. 22, 4-16 (1934; Zbl 0008.14903)]; and also it is known that for a typical continuous function \(J_ f\) is of the 1st category, even \(\sigma\)-porous [the second author, Real Anal. Exch. 13(1987/88), 314-350 (1988; Zbl 0666.26003)].

The present paper contains two main results: (I) If \[ f(x)= \sum_{n\geq 1}(n!)^{-2}\sin [2\pi((n+ 2)!)^ 5x] \] then \(J_ f= \mathbb{R}\); (II) For typically continuous functions \(J_ f\) is \({\mathbf c}\)-dense, that is, \(J_ f\) has cardinality \({\mathbf c}\) in every interval. In the case of (I) this gives an explicit construction of a function \(f\) with \(J_ f= \mathbb{R}\); and in fact a stronger result is proved for which reference to the paper should be made. The proof of (II) is similar to a proof of S. Saks [Fundam. Math. 19, 211-219 (1932; Zbl 0005.39105)], relying on the Banach-Mazur game method, and an important lemma by the authors, Lemma 2, closely connected to the classical condition (D) [S. Saks, Theory of the integral, 2nd ed. (1937; Zbl 0017.30004); p. 290, 9.2 Lemma].

The present paper contains two main results: (I) If \[ f(x)= \sum_{n\geq 1}(n!)^{-2}\sin [2\pi((n+ 2)!)^ 5x] \] then \(J_ f= \mathbb{R}\); (II) For typically continuous functions \(J_ f\) is \({\mathbf c}\)-dense, that is, \(J_ f\) has cardinality \({\mathbf c}\) in every interval. In the case of (I) this gives an explicit construction of a function \(f\) with \(J_ f= \mathbb{R}\); and in fact a stronger result is proved for which reference to the paper should be made. The proof of (II) is similar to a proof of S. Saks [Fundam. Math. 19, 211-219 (1932; Zbl 0005.39105)], relying on the Banach-Mazur game method, and an important lemma by the authors, Lemma 2, closely connected to the classical condition (D) [S. Saks, Theory of the integral, 2nd ed. (1937; Zbl 0017.30004); p. 290, 9.2 Lemma].

Reviewer: P.S.Bullen (Vancouver)