Micro tangent sets of continuous functions

Zoltán Buczolich

Mathematica Bohemica (2003)

  • Volume: 128, Issue: 2, page 147-167
  • ISSN: 0862-7959

Abstract

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Motivated by the concept of tangent measures and by H. Fürstenberg’s definition of microsets of a compact set A we introduce micro tangent sets and central micro tangent sets of continuous functions. It turns out that the typical continuous function has a rich (universal) micro tangent set structure at many points. The Brownian motion, on the other hand, with probability one does not have graph like, or central graph like micro tangent sets at all. Finally we show that at almost all points Takagi’s function is graph like, and Weierstrass’s nowhere differentiable function is central graph like.

How to cite

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Buczolich, Zoltán. "Micro tangent sets of continuous functions." Mathematica Bohemica 128.2 (2003): 147-167. <http://eudml.org/doc/249212>.

@article{Buczolich2003,
abstract = {Motivated by the concept of tangent measures and by H. Fürstenberg’s definition of microsets of a compact set $A$ we introduce micro tangent sets and central micro tangent sets of continuous functions. It turns out that the typical continuous function has a rich (universal) micro tangent set structure at many points. The Brownian motion, on the other hand, with probability one does not have graph like, or central graph like micro tangent sets at all. Finally we show that at almost all points Takagi’s function is graph like, and Weierstrass’s nowhere differentiable function is central graph like.},
author = {Buczolich, Zoltán},
journal = {Mathematica Bohemica},
keywords = {typical continuous function; Brownian motion; Takagi’s function; Weierstrass’s function; typical continuous function; Brownian motion; Takagi function; Weierstrass function},
language = {eng},
number = {2},
pages = {147-167},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Micro tangent sets of continuous functions},
url = {http://eudml.org/doc/249212},
volume = {128},
year = {2003},
}

TY - JOUR
AU - Buczolich, Zoltán
TI - Micro tangent sets of continuous functions
JO - Mathematica Bohemica
PY - 2003
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 128
IS - 2
SP - 147
EP - 167
AB - Motivated by the concept of tangent measures and by H. Fürstenberg’s definition of microsets of a compact set $A$ we introduce micro tangent sets and central micro tangent sets of continuous functions. It turns out that the typical continuous function has a rich (universal) micro tangent set structure at many points. The Brownian motion, on the other hand, with probability one does not have graph like, or central graph like micro tangent sets at all. Finally we show that at almost all points Takagi’s function is graph like, and Weierstrass’s nowhere differentiable function is central graph like.
LA - eng
KW - typical continuous function; Brownian motion; Takagi’s function; Weierstrass’s function; typical continuous function; Brownian motion; Takagi function; Weierstrass function
UR - http://eudml.org/doc/249212
ER -

References

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