  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.
