Approximate differentiation: Jarník points

• Autorzy: Jan Malý , Luděk Zajíček
• Języki publikacji: EN

Abstrakty

EN

We investigate Jarník's points for a real function f defined in ℝ, i.e. points x for which $ap_{y → x}|(f(y)-f(x))/(y-x)|=+∞$. In 1970, Berman has proved that the set $J_f$ of all Jarník's points for a path f of the one-dimensional Brownian motion is the whole ℝ almost surely. We give a simple explicit construction of a continuous function f with $J_f = ℝ. The main result of our paper says that for a typical continuous function f on [0,1] the set $J_f$ is c-dense in [0,1].

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Fundamenta Mathematicae
• Rocznik: 1991-1992
• Tom: 140
• Numer: 1
• Strony: 87-97

Daty

wydano

1991

otrzymano

1991-04-22

Twórcy

autor

Jan Malý

• Faculty of Mathematics and Physics (KMA), Charles University, Sokolovská 83, 18600 Praha 8, Czechoslovakia

autor

Luděk Zajíček

• Faculty of Mathematics and Physics (KMA), Charles University, Sokolovská 83, 18600 Praha 8, Czechoslovakia

Bibliografia

• [1] S. M. Berman, Gaussian processes with stationary increments: Local times and sample function properties, Ann. Math. Statist. 41 (1970), 1260-1272.
• [2] D. Geman and J. Horowitz, Occupation densities, Ann. Probab. 8 (1980), 1-67.
• [3] M. de Guzmán, A general form of the Vitali theorem, Colloq. Math. 34 (1975), 69-72.
• [4] V. Jarník, Sur les nombres dérivées approximatifs, Fund. Math. 22 (1934), 4-16.
• [5] J. C. Oxtoby, The Banach-Mazur game and Banach category theorem, in: Contribution to the Theory of Games III, Ann. of Math. Stud. 39, Princeton 1957, 159-163.
• [6] J. C. Oxtoby, Measure and Category, Springer, New York 1980.
• [7] S. Saks, On the functions of Besicovitch in the space of continuous functions, Fund. Math. 19 (1932), 211-219.
• [8] S. Saks, Theory of the Integral, Monograf. Mat. 7, Warszawa 1937 (reprinted by Hafner Publ., New York).
• [9] L. Zajíček, The differentiability structure of typical functions in C[0,1]), Real Anal. Exchange 13 (1987-88), 119, 103-106, 93.
• [10] L. Zajíček, Porosity, derived numbers and knot points of typical continuous functions, Czechoslovak Math. J. 39 (114) (1989), 45-52.