Level Sets of the Takagi Function: Local Level Sets

Details Level Sets of the Takagi Function: Local Level Sets Level Sets of the Takagi Function: Local Level Sets

2010-09-04

arxiv.org/abs/1009.0855v7

Monat. Math. 2012

Mathematics

Classical Analysis and ODEs

32 pages, 2 figures, 1 table. Latest version has updated equation numbering. The final publication will soon be available at s

Scientific paper

10.1007/s00605-012-0399-4

The Takagi function \tau : [0, 1] \to [0, 1] is a continuous non-differentiable function constructed by Takagi in 1903. The level sets L(y) = {x : \tau(x) = y} of the Takagi function \tau(x) are studied by introducing a notion of local level set into which level sets are partitioned. Local level sets are simple to analyze, reducing questions to understanding the relation of level sets to local level sets, which is more complicated. It is known that for a "generic" full Lebesgue measure set of ordinates y, the level sets are finite sets. Here it is shown for a "generic" full Lebesgue measure set of abscissas x, the level set L(\tau(x)) is uncountable. An interesting singular monotone function is constructed, associated to local level sets, and is used to show the expected number of local level sets at a random level y is exactly 3/2.

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