Geodesic
Approximate differentiation: Jarník points
Jan Malý1 ; Luděk Zajı́ček1
1Faculty of Mathematics and Physics (KMA) Charles Yniversity Sokolovská 83 18600 Praha 8, Czechoslovakia
Fundamenta Mathematicae, Tome 140 (1991) no. 1, pp. 87-97
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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].
DOI : 10.4064/fm-140-1-87-97

Jan Malý  1   ; Luděk Zajı́ček  1

1 Faculty of Mathematics and Physics (KMA) Charles Yniversity Sokolovská 83 18600 Praha 8, Czechoslovakia 
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Jan Malý; Luděk Zajı́ček. Approximate differentiation: Jarník points. Fundamenta Mathematicae, Tome 140 (1991) no. 1, pp. 87-97. doi: 10.4064/fm-140-1-87-97

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