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Let T be Takagi's continuous but nowhere-differentiable function. It is knownthat almost all level sets (with respect to Lebesgue measure on the range of T)are finite. We show that the most common cardinality of the level sets of T istwo, and investigate in detail the set of ordinates y such that the level setat level y has precisely two elements. As a by-product, we obtain a simpleiterative procedure for solving the equation T(x)=y. We show further that anypositive even integer occurs as the cardinality of some level set, andinvestigate which cardinalities occur with positive probability if an ordinatey is chosen at random from the range of T. The key to the results is a systemof set equations for the level sets, which are derived from the partialself-similarity of T. These set equations yield a system of linearrelationships between the cardinalities of level sets at various levels, fromwhich all the results of this paper flow.

On the distribution of the cardinalities of level sets of the Takagi function