  This paper examines level sets of functions of the form
$f(x)=\sum_{n=0}^\infty \frac{r_n}{2^n}\phi(2^n x)$, where phi(x) is the
distance from x to the nearest integer, and r_n equals 1 or -1 for each n. Such
functions are referred to as signed Takagi functions. The case when r_n=1 for
all n is the classical Takagi function, a well-known example of a continuous
but nowhere differentiable function. For f of the above form, the maximum and
minimum values of f are expressed in terms of the sequence {r_n}. It is then
shown that almost all level sets of f are finite (with respect to Lebesgue
measure on the range of f), but the set of ordinates y with an uncountably
large level set is residual in the range of f. The concept of a local level set
of the Takagi function, due to Lagarias and Maddock, is extended to arbitrary
signed Takagi functions. It is shown that the average number of local level
sets contained in a level set of f is the reciprocal of the height of the graph
of f, and consequently, this average lies between 3/2 and 2.
