Approximate differentiation: Jarník points

Tom 140 / 1991

Jan Malý, Luděk Zajı́ček Fundamenta Mathematicae 140 (1991), 87-97 DOI: 10.4064/fm-140-1-87-97

Streszczenie

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].

Autorzy

  • Jan MalýFaculty of Mathematics and Physics (KMA)
    Charles Yniversity
    Sokolovská 83
    18600 Praha 8, Czechoslovakia
  • Luděk Zajı́čekFaculty of Mathematics and Physics (KMA)
    Charles Yniversity
    Sokolovská 83
    18600 Praha 8, Czechoslovakia

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