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Approximate differentiation: Jarník points. (English) Zbl 0814.26005
If $$f: \mathbb{R}\to \mathbb{R}$$ 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].

##### MSC:
 26A24 Differentiation (real functions of one variable): general theory, generalized derivatives, mean value theorems 26A27 Nondifferentiability (nondifferentiable functions, points of nondifferentiability), discontinuous derivatives
##### Keywords:
approximate differentiation; Jarník points
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