  This paper examines level sets of two families of continuous, nowhere
differentiable functions (one a subfamily of the other) defined in terms of the
"tent map". The well-known Takagi function is a special case. Sharp upper
bounds are given for the Hausdorff dimension of the level sets of functions in
these two families. Furthermore, the case where a function f is chosen at
random from either family is considered, and results are given for the
Hausdorff dimension of the zero set and the set of maximum points of f.
