  This paper examines the level sets of the continuous but nowhere
differentiable functions \begin{equation*} f_r(x)=\sum_{n=0}^\infty
r^{-n}\phi(r^n x), \end{equation*} where $\phi(x)$ is the distance from $x$ to
the nearest integer, and $r$ is an integer with $r\geq 2$. It is shown, by
using properties of a symmetric correlated random walk, that almost all level
sets of $f_r$ are finite (with respect to Lebesgue measure on the range of
$f$), but that for an abscissa $x$ chosen at random from $[0,1)$, the level set
at level $y=f_r(x)$ is uncountable almost surely. As a result, the occupation
measure of $f_r$ is singular.
