MATHEMATICA BOHEMICA, Vol. 128, No. 2, pp. 147-167 (2003)

Micro tangent sets of continuous functions

Zoltan Buczolich

Zoltan Buczolich, Department of Analysis, Eötvös Lorand, University, Pazmany Péter Sétany 1/c, 1117 Budapest, Hungary, e-mail:, {\tt {}buczo}

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.

Keywords: typical continuous function, Brownian motion, Takagi's function, Weierstrass's function

Classification (MSC2000): 26A27, 26A24, 28A78, 60J65

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