Level sets of the Takagi function: local level sets

Abstract

The Takagi function τ: [0,1] → [0, 1] is a continuous non-differentiable function constructed by Takagi in 1903. The level sets L(y) = {x : τ(x) = y} of the Takagi function τ(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. In contrast, here it is shown for a “generic” full Lebesgue measure set of abscissas x, the level set L(τ(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 \({\frac{3}{2}}\).